MULTIGRID SOLUTION OF THEEULER EQUATIONS WITH LOCALPRECONDITIONINGbyJohn Francis LynnA dissertation submitted in partial ful�llmentof the requirements for the degree ofDoctor of Philosophy(Aerospace Engineering and Scienti�c Computing)in The University of Michigan1995Doctoral Committee:Professor Bram van Leer, ChairpersonProfessor Edward LarsenAssociate Professor Kenneth G. PowellProfessor Philip L. Roe

To my beloved wife Nayana, who has been my supportthrough the ups and downs of my life as a doctoral student.

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ACKNOWLEDGEMENTSI would like to thank my chairperson, Professor Bram van Leer, for the role hehas played as my graduate advisor. His suggestions have always been excellent andhe has been a pleasure to work with. I would also like to thank Professors Ken Powelland Phil Roe for being there whenever I needed any help.I am grateful to Professor Edward Larsen of Nuclear Engineering for agreeingto be a member of my doctoral committee and for his comments regarding thisdissertation. I would also like to acknowledge the suggestions made by Dr. SteveAllmaras of Boeing with regard to this work. He has made many helpful suggestionsabout my work and, in particular, this dissertation.I have come to know many of the other members of the W. M. Keck CFD lab-oratory quite well and would like to acknowledge the positive e�ect that this grouphas had on my research. I feel truly fortunate to have been a member of this group.Dohyung Lee and Dave Darmofal deserve special mention for the many useful dis-cussions that we had on local preconditioning as well as on multigrid methods.This work has been funded in part by the Boeing Commercial Airplane Companyand the Air Force O�ce of Scienti�c Research.iii

TABLE OF CONTENTSDEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iiiLIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viiLIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xivCHAPTERI. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : 1II. THE MULTIGRID METHOD : : : : : : : : : : : : : : : : : : : 62.1 Elements of Linear Multigrid : : : : : : : : : : : : : : : : : : 72.1.1 Obtaining an initial guess with nested iteration : : : 102.1.2 The Coarse-Grid Correction (CGC) scheme : : : : : 112.1.3 Two-level analysis in Fourier space : : : : : : : : : : 122.1.4 The V-Cycle and the �-Cycle : : : : : : : : : : : : : 142.2 Aspects of Nonlinear Multigrid : : : : : : : : : : : : : : : : : 162.2.1 Moving from CGC to FAS : : : : : : : : : : : : : : 162.2.2 FAS with nonlinear problems : : : : : : : : : : : : : 182.2.3 Nonlinear versions of the V and � cycles : : : : : : 192.2.4 Choice of restriction and prolongation operators : : 212.3 A Nonlinear Variant - Semi-Coarsened Multigrid : : : : : : : 222.3.1 Mulder's semi-coarsened multigrid method : : : : : 242.4 The Defect-Correction Technique : : : : : : : : : : : : : : : : 282.5 Full Multigrid (FMG) : : : : : : : : : : : : : : : : : : : : : : 292.6 Elements of the Update Scheme : : : : : : : : : : : : : : : : 312.6.1 Spatial discretization : : : : : : : : : : : : : : : : : 322.6.2 Boundary conditions : : : : : : : : : : : : : : : : : 372.6.3 Temporal discretization : : : : : : : : : : : : : : : : 392.7 Implications for Parallel Processing : : : : : : : : : : : : : : : 40III. LOCAL PRECONDITIONING : : : : : : : : : : : : : : : : : : 42iv

3.1 Three Ways of Time Stepping : : : : : : : : : : : : : : : : : : 433.2 Preconditioning the One-Dimensional Euler Equations : : : : 443.3 Two-Dimensional Preconditioning : : : : : : : : : : : : : : : 463.4 Three-Dimensional Preconditioning : : : : : : : : : : : : : : : 483.5 Implementing Characteristic Time-Stepping : : : : : : : : : : 493.6 Reformulation of Numerical Fluxes : : : : : : : : : : : : : : : 513.6.1 Roe's arti�cial-viscosity matrix : : : : : : : : : : : : 523.6.2 Modi�ed Roe arti�cial-viscosity matrix : : : : : : : 533.7 Fourier Footprints of the Spatial Euler Operator : : : : : : : 553.7.1 Fourier footprint of the �rst-order upwind scheme : 563.7.2 Fourier footprints of the � family of upwind schemes 573.8 E�ect of Preconditioning on the Spatial Operator : : : : : : : 573.9 Relocating High-Frequency Content in the Footprint : : : : : 633.10 E�ect of the Flow Angle and Aspect Ratio on the Footprint : 66IV. DESIGN OF OPTIMAL MULTI-STAGE SCHEMES : : : : 684.1 Previous Scalar Analysis : : : : : : : : : : : : : : : : : : : : : 694.2 Extensions of Scalar Analysis : : : : : : : : : : : : : : : : : : 724.3 Formulation of the Optimization Problem : : : : : : : : : : : 754.4 Dependence on the Flow Angle : : : : : : : : : : : : : : : : : 784.4.1 Obtaining a de�nition for the Courant number length-scale : : : : : : : : : : : : : : : : : : : : : : : : : : 804.5 Optimization over Entire High-Frequency Domain : : : : : : 814.5.1 Dependence on Mach Number : : : : : : : : : : : : 874.5.2 Optimal multi-stage coe�cients : : : : : : : : : : : 894.6 Optimization over High-High Frequency Domain : : : : : : : 994.6.1 Dependence on Mach number : : : : : : : : : : : : 1014.6.2 Optimal multi-stage coe�cients : : : : : : : : : : : 102V. FURTHER ANALYSIS : : : : : : : : : : : : : : : : : : : : : : : 1115.1 De�ning the Courant Number Length-Scale for an ArbitraryQuadrilateral Cell : : : : : : : : : : : : : : : : : : : : : : : : 1115.2 Multi-Stage Coe�cients for Navier-Stokes Spatial Operators : 1135.2.1 Reformulating the optimization problem : : : : : : 1145.2.2 Multi-stage coe�cients with prescribed damping forNavier-Stokes operators : : : : : : : : : : : : : : : : 1205.3 Discretizations Incorporating Explicit Residual Smoothing : : 1215.3.1 Choice of smoothing stencil : : : : : : : : : : : : : : 1225.3.2 Optimal multi-stage coe�cients : : : : : : : : : : : 1235.4 Optimal Schemes for Three-Dimensional Euler Operators : : 1265.5 Discretizations on Unstructured Meshes : : : : : : : : : : : : 1355.5.1 Discretizations on triangular meshes : : : : : : : : : 136v

5.5.2 Discretizations on Cartesian-unstructured meshes : 138VI. NUMERICAL STUDIES : : : : : : : : : : : : : : : : : : : : : : 1396.1 Ways of Implementing Local Preconditioning withMulti-StageTime-Stepping : : : : : : : : : : : : : : : : : : : : : : : : : : 1406.2 Smoothing about a Sonic Point : : : : : : : : : : : : : : : : : 1416.3 Smoothing about a Stagnation Point : : : : : : : : : : : : : : 1436.4 Test Problem: Random Perturbations on a Square Mesh : : : 1436.4.1 Grid-convergence studies : : : : : : : : : : : : : : : 1466.4.2 Importance of the optimal multi-stage coe�cients : 1516.4.3 Comparison of damping rates per multigrid cycle : : 1526.5 Test Problem: Flow past a Bump in a Channel : : : : : : : : 1536.5.1 Subsonic case : : : : : : : : : : : : : : : : : : : : : 1556.5.2 Transonic case : : : : : : : : : : : : : : : : : : : : : 1576.5.3 Supersonic case : : : : : : : : : : : : : : : : : : : : 1636.5.4 Summary of results - channel problem : : : : : : : : 1666.6 Test Problem: Transonic Flow past a Bump on a Wall in aSemi-in�nite Domain : : : : : : : : : : : : : : : : : : : : : : 1686.7 Improving Robustness with Explicit Residual Smoothing : : : 170VII. CONCLUSIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1727.1 Future Work : : : : : : : : : : : : : : : : : : : : : : : : : : : 173APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 177BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 189vi

LIST OF FIGURESFigure2.1 Discrete representation of a sinusoidal wave-mode (k = 6) on threedi�erent grid levels : : : : : : : : : : : : : : : : : : : : : : : : : : : 92.2 Schedule of grids for (a) V-cycle, (b) W-cycle and (c) Sawtooth cycle,all on four levels : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 142.3 Arrangement of �nest (8 � 8) and coarser grids for problems withalignment. The arrows indicate how the grids are linked by restric-tion (downward) and prolongation (upward). : : : : : : : : : : : : : 252.4 Full Multigrid cycle incorporating a single (�3 = 1) Sawtooth-FAScycle on each intermediate grid level : : : : : : : : : : : : : : : : : : 302.5 Obtaining the boundary-interface ux with the ghost-cell approach:the ux is obtained by solving the associated Riemann problem withthe states in the ghost cell and boundary cell as initial states. : : : : 383.1 Fourier footprint of the �rst-order upwind approximation of the spa-tial Euler operator, for M = 0:1, and ow angle � = 0�. Thetime-step chosen corresponds to a Courant-number value of 1. : : : 583.2 Fourier footprint of the �rst-order upwind approximation of the spa-tial Euler operator, for M = 0:5, and ow angle � = 0�. : : : : : : 593.3 Fourier footprint of the �rst-order upwind approximation of the spa-tial Euler operator, for M = 0:9, and ow angle � = 0�. : : : : : : 593.4 Fourier footprint of the �rst-order upwind approximation of the spa-tial Euler operator, for M = 2, and ow speed aligned with thegrid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 603.5 Fourier footprint of the preconditioned �rst-order upwind approxi-mation of the spatial Euler operator, for M = 0:1, and ow angle� = 0�. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60vii

3.6 Fourier footprint of the preconditioned �rst-order upwind approxi-mation of the spatial Euler operator, for M = 0:5, and ow angle� = 0�. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 613.7 Fourier footprint of the preconditioned �rst-order upwind approxi-mation of the spatial Euler operator, for M = 0:9, and ow angle� = 0�. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 613.8 Fourier footprint of the preconditioned �rst-order upwind approx-imation of the spatial Euler operator, for M = 2, and ow angle� = 0�. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 623.9 Fourier footprint of the preconditioned �rst-order upwind approxi-mation of the spatial Euler operator, for M = 0:1, and ow angle� = 0�. j�xj; j�yj 2 [0; �] : Courant number = 2. Modi�ed precondi-tioner (Section 3.9) used here. : : : : : : : : : : : : : : : : : : : : : 633.10 Fourier footprint of the preconditioned �rst-order upwind approxi-mation of the spatial Euler operator, for M = 0:1, and ow angle� = 45�. j�xj; j�yj 2 [0; �] : Courant number = 2. Modi�ed precon-ditioner (Section 3.9) used here. : : : : : : : : : : : : : : : : : : : 663.11 Fourier footprint of the preconditioned �rst-order upwind approxi-mation of the spatial Euler operator, for M = 0:1, and ow angle� = 0�. AR = 4. j�xj; j�yj 2 [0; �] : Courant number = 2. Modi�edpreconditioner (Section 3.9) used here. : : : : : : : : : : : : : : : : 674.1 Fourier footprint (dashed line) of the third-order upwind-biased spa-tial discretization of the one-dimensional convection operator, andlevel lines (solid) of the ampli�cation factor of Tai's optimal six-stagescheme. (Design-graph of Tai's optimal six-stage scheme). : : : : : 704.2 Modulus of the ampli�cation factor as a function of spatial frequency,for the case of Figure 4.1 . : : : : : : : : : : : : : : : : : : : : : : : 714.3 Fourier footprint of the �rst-order upwind approximation of the two-dimensional convection operator; convection angle � = 10�. : : : : 724.4 Fourier footprint of the �rst-order upwind approximation of the two-dimensional convection equation; convection angle � = 45�. : : : : 734.5 High-frequency Fourier footprint of the �rst-order upwind 2-D scalaradvection operator plotted on top of the level lines of the ampli�ca-tion factor of the 4-stage scheme optimized using the modi�ed Roeoperator for the Euler equations. Convection angle 45�. : : : : : : : 74viii

4.6 Wedge-�lter that is used to make the optimization problemmeaning-ful for ow-angles near 0 (or �=2). The shaded region represents theportion of the high-frequency domain that is considered; its bound-ary is controlled by a \wedge" drawn in the central, low-frequencydomain. In this case, � < �=4 and the wedge angle is given by = �+�=4. For �=4 � � � �=2, the shaded region would rotate by�=2, such that high-frequency combinations around the �x axis areexcluded. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 764.7 Variation of the optimal Courant number with the ow angle, for the�rst-order upwind 4-stage scheme. Optimization over entire high-frequency range minus region removed by wedge �lter (see Section4.3). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 794.8 A geometrical interpretation of Equation (4.11 ). l � AE is thedistance from a cell corner to the opposite cell-diagonal. : : : : : : : 804.9 Design-graph for the optimal 3-stage scheme based on the precondi-tioned �rst-order upwind Euler operator. Flow angle 45�. : : : : : : 834.10 As Figure 4.9 , but for optimal 4-stage scheme. : : : : : : : : : : : : 834.11 As Figure 4.9 , but for optimal 5-stage scheme. : : : : : : : : : : : : 844.12 As Figure 4.10 , but for ow angle 0�. : : : : : : : : : : : : : : : : : 844.13 Level lines of the maximum ampli�cation factor in the (�x; �y) planeover the high-frequency domain. 1st order 4-stage scheme, � = 45�. 854.14 As Figure 4.13 , but for acoustic waves only. : : : : : : : : : : : : : 854.15 As Figure 4.13 , but for shear/entropy waves only. : : : : : : : : : : 864.16 As Figure 4.15 , but for ow angle 0�. The dashed line outlines theportion of the domain considered (the portion containing the line�y = 0). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 864.17 Design-graph of the optimal 4-stage scheme based on the precondi-tioned �rst-order upwind Euler operator. Flow angle 0�, M = 0:5. : 874.18 Design-graph of the optimal 4-stage scheme based on the precondi-tioned �rst-order upwind Euler operator. Flow angle 0�, M = 0:9. : 884.19 As Figure 4.12 but for � = 0. : : : : : : : : : : : : : : : : : : : : : : 90ix

4.20 As Figure 4.12 but for � = �1. : : : : : : : : : : : : : : : : : : : : : 904.21 As Figure 4.12 but for � = 1=3. : : : : : : : : : : : : : : : : : : : : 914.22 Sub-optimal 4-stage scheme obtained in [34] . M = 0:1, � = 45�.�1 = 0:1624, �2 = 0:2755, �3 = 0:5025, � = 1:6073, �opt = 0:6641 : : 914.23 Variation with ow angle of multi-stage coe�cients and Courantnumber (based on a �xed length scale independent of ow angle) fora �rst-order 4-stage optimal scheme. Optimization over high-highfrequency domain only. : : : : : : : : : : : : : : : : : : : : : : : : : 994.24 Fourier footprint of the �rst-order upwind approximation of the spa-tial Euler operator with the preconditioner of Van Leer et al. forM = 0:1, and ow angle � = 0�. Footprint corresponds to high-highfrequency domain. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1004.25 Design-graph of optimal �rst-order upwind four-stage scheme ob-tained by optimizing over its high-high frequency footprint. M =0:1, � = 0. �opt = 0:0632. : : : : : : : : : : : : : : : : : : : : : : : : 1014.26 Variation of multi-stage coe�cients and Courant number with Machnumber for a �rst-order 4-stage optimal scheme. Optimization overhigh-high frequency domain only. : : : : : : : : : : : : : : : : : : : 1024.27 As Figure 4.25 but for � = 0. : : : : : : : : : : : : : : : : : : : : : : 1034.28 As Figure 4.25 but for � = �1. : : : : : : : : : : : : : : : : : : : : : 1044.29 As Figure 4.25 but for � = 1=3. : : : : : : : : : : : : : : : : : : : : 1044.30 Design graph of optimal two-stage scheme based on high-high fre-quency domain together with Fourier footprint of entire spatial oper-ator. � = �1 spatial discretization. There is a small transgression ofthe footprint across the stability boundary near the origin (invisiblehere). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1055.1 Quadrilateral cell with equivalent rotated rectangular cell in the owdirection. The cell has an equivalent aspect-ratio ARq = �x1=�y1. 1125.2 Four-stage scheme obtained by optimizing over high-high frequencyfootprint of the discrete, preconditioned Navier-Stokes operator. �minwas prescribed as 0.1. M = 0:1, � = 0, Re�x = 0:1, AR = 1. � =7.891. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 116x

5.3 Optimal four-stage scheme obtained by optimizing over high-highfrequency footprint of the discrete, preconditioned Navier-Stokes op-erator. M = 0:1, � = 0, Re�x = 0:1, AR = 1. �opt = 0.0034, � =2.953. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1175.4 Four-stage scheme obtained by optimizing over high-high frequencyfootprint of the discrete, preconditioned Navier-Stokes operator. �minwas prescribed as 0.2. M = 0:1, � = 0, Re�x = 0:1, AR = 1. �= 12.845. This scheme will be unstable for some \high-low", \low-high" and \low-low" frequency combinations. : : : : : : : : : : : : 1175.5 Four-stage scheme obtained by optimizing over high-high frequencyfootprint of the discrete, preconditioned Navier-Stokes operator witha constraint on Pmax introduced via a penalty function. �min wasprescribed as 0.2. M = 0:1, � = 0, Re�x = 0:1, AR = 1. � = 11.36. 1195.6 Four-stage scheme obtained by optimizing over high-high frequencyfootprint of the discrete, preconditioned Navier-Stokes operator witha constraint on P �max (Equation 5.10 ) introduced via a penalty func-tion. �min was prescribed as 0.2. M = 0:1, � = 0, Re�x = 0:1, AR= 1. � = 0:2. � = 9.5361. : : : : : : : : : : : : : : : : : : : : : : : : 1195.7 Variation with cell-Reynolds number of multistage coe�cients andCourant number for a �rst-order 4-stage scheme with prescribeddamping. Optimization based high-high frequency footprint of thediscrete, preconditioned Navier-Stokes operator with a constraint onPmax introduced via a penalty function. �min was prescribed as 0.15.M = 0:1, � = 0, AR = 1, � = 0:05. : : : : : : : : : : : : : : : : : : 1215.8 Fourier footprint of the �rst-order preconditioned Euler operatorwith explicit residual smoothing. Regular Laplacian stencil usedfor smoothing. M = 0:1, � = 45�, � = 0:1. : : : : : : : : : : : : : : 1245.9 Fourier footprint of the �rst-order preconditioned Euler operatorwith explicit residual smoothing. Rotated Laplacian stencil usedfor smoothing. M = 0:1, � = 45�, � = 0:1. : : : : : : : : : : : : : : 1245.10 Four-stage scheme obtained by optimizing over footprint of the �rst-order preconditioned Euler operator with explicit residual smooth-ing. Regular Laplacian stencil used for smoothing. M = 0:1,� = 45�, � = 0:1, � = 3:25. : : : : : : : : : : : : : : : : : : : : : : : 125xi

5.11 Optimal four-stage scheme based on the Fourier footprint of the k =0 cell-vertex scheme for the 2-D convection equation on a triangularmesh. � = 30�, �1 = 0:07051, �2 = 0:1803, �3 = 0:3854, � = 3:1079,�opt = 0:3559. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1366.1 Sonic-point smoothing function for �. : : : : : : : : : : : : : : : : 1426.2 Residual-history plots. M = 0:35, � = 10�. First-order spatialoperator. MG-MTS with four-stage time-stepping scheme and 4� 4grid as coarsest grid level. Periodic boundary conditions. : : : : : : 1476.3 Residual-history plots. As Figure 6.2 but for M = 0:85. : : : : : : : 1486.4 Residual-history plots. As Figure 6.2 but for M = 2:0. : : : : : : : : 1486.5 Residual-history plots. M = 0:35, � = 10�. First order spatialoperator. MG-MTS with four-stage time-stepping scheme and 4� 4grid as coarsest grid level. In ow-out ow boundary conditions. : : : 1496.6 Residual-history plots. As Figure 6.5 but for M = 0:85. : : : : : : : 1506.7 Residual-history plots. As Figure 6.5 but for M = 2:0. : : : : : : : : 1506.8 Residual-history plots for the transonic channel case (M = 0:85).64x32 grid. First-order spatial di�erencing. Four-stage time-steppingschemes; 8x4 grid taken as coarsest grid for multigrid. The conver-gence history for a single-grid with local time-stepping is not shown;it lies almost on top of the curve for the single-grid preconditionedscheme. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1566.9 Mach number contours. Flow past a non-smooth bump in a channel.M = 0:35, t=c = 0:042, 128 � 64 grid, spatial discretization with� = 0, Van Albada's limiter and modi�ed Roe ux. : : : : : : : : : 1586.10 Flow past a non-smooth bump in a channel. M = 0:85, t=c = 0:042,128 � 64 grid. Spatial di�erencing with � = 0, Van Albada's limiterand modi�ed Roe ux. : : : : : : : : : : : : : : : : : : : : : : : : : 1616.11 Flow past a non-smooth bump in a channel. M = 1:4, t=c = 0:042,128�64 grid. Spatial discretization with � = 0, Van Albada's limiterand modi�ed Roe ux. : : : : : : : : : : : : : : : : : : : : : : : : : 1636.12 Flow past a non-smooth bump in a channel. M = 1:4, t=c = 0:042,64�32 grid. Spatial discretization with � = 0, Van Albada's limiterand modi�ed Roe ux. : : : : : : : : : : : : : : : : : : : : : : : : : 164xii

6.13 Transonic ow past a non-smooth bump on a wall in a semi-in�nitedomain. M = 0:85, t=c = 0:042, 128�64 grid. Spatial discretizationwith � = 0, Van Albada's limiter and modi�ed Roe scheme. : : : : : 1686.14 Flow past a non-smooth bump in a channel. M = 0:99, t=c = 0:042,128 � 64 grid. Spatial discretization with � = 0, Van Albada'slimiter and modi�ed Roe ux. Explicit residual-smoothing was usedto obtain this solution. : : : : : : : : : : : : : : : : : : : : : : : : : 170A.1 Design-graph of multi-stage coe�cients used as an initial guess. First-order preconditioned spatial operator with M = 0:1, � = 0�.�1 =0:1, �2 = 0:2, �3 = 0:35 and � = 1:5. � = 0:4164. : : : : : : : : : : : 185A.2 Temperature schedule that was used to solve the optimization prob-lem. The temperature was reduced at each step by using a multi-plicative factor of 0.8. : : : : : : : : : : : : : : : : : : : : : : : : : : 186A.3 Design-graph of optimal multi-stage coe�cients. First-order pre-conditioned spatial operator with M = 0:1, � = 0�.�1 = 0:07876,�2 = 0:2004, �3 = 0:4241 and � = 2:63. � = 0:2349 : : : : : : : : : 187A.4 Convergence histories of the parameters involved in the optimization.Also plotted are the values of � at each step. : : : : : : : : : : : : : 187

xiii

LIST OF TABLESTable4.1 Optimal multi-stage coe�cients for �rst-order scheme. Optimizationbased on entire high-frequency domain minus �ltered region (seeSection 4.3 ). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 954.2 Optimal multi-stage coe�cients for � = 0 scheme. Optimizationbased on entire high-frequency domain minus �ltered region (seeSection 4.3 ). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 964.3 Optimal multi-stage coe�cients for � = �1 scheme. Optimizationbased on entire high-frequency domain minus �ltered region (seeSection 4.3 ). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 974.4 Optimal multi-stage coe�cients for � = 1=3 scheme. Optimizationbased on entire high-frequency domain minus �ltered region (seeSection 4.3 ). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 984.5 Optimal multi-stage coe�cients for �rst-order scheme. Optimizationbased on high-high frequency domain. : : : : : : : : : : : : : : : : : 1074.6 Optimal multi-stage coe�cients for � = 0 scheme. Optimizationbased on high-high frequency domain. : : : : : : : : : : : : : : : : : 1084.7 Optimal multi-stage coe�cients for � = �1 scheme. Optimizationbased on high-high frequency domain. : : : : : : : : : : : : : : : : : 1094.8 Optimal multi-stage coe�cients for � = 1=3 scheme. Optimizationbased on high-high frequency domain. : : : : : : : : : : : : : : : : : 1105.1 Optimal multi-stage coe�cients for �rst-order scheme with explicitresidual smoothing, � = 0:1. Optimization based on entire high-frequency domain minus �ltered region (see Section 4.3 ). : : : : : : 1275.2 Optimal multi-stage coe�cients for � = 0 scheme with explicit resid-ual smoothing, � = 0:1. Optimization based on entire high-frequencydomain minus �ltered region (see Section 4.3 ). : : : : : : : : : : : : 128xiv

5.3 Optimal multi-stage coe�cients for � = �1 scheme with explicitresidual smoothing, � = 0:1. Optimization based on entire high-frequency domain minus �ltered region (see Section 4.3 ). : : : : : : 1295.4 Optimal multi-stage coe�cients for � = 1=3 scheme with explicitresidual smoothing, � = 0:1. Optimization based on entire high-frequency domain minus �ltered region (see Section 4.3 ). : : : : : : 1305.5 Optimal multi-stage coe�cients for �rst-order scheme with explicitresidual smoothing, � = 0:1. Constrained optimization based onhigh-high frequency domain. : : : : : : : : : : : : : : : : : : : : : : 1315.6 Optimal multi-stage coe�cients for � = 0 scheme with explicit resid-ual smoothing, � = 0:1. Constrained optimization based on high-high frequency domain. : : : : : : : : : : : : : : : : : : : : : : : : : 1325.7 Optimal multi-stage coe�cients for � = �1 scheme with explicitresidual smoothing, � = 0:1. Constrained optimization based onhigh-high frequency domain. : : : : : : : : : : : : : : : : : : : : : : 1335.8 Optimal multi-stage coe�cients for � = 1=3 scheme with explicitresidual smoothing, � = 0:1. Constrained optimization based onhigh-high frequency domain. : : : : : : : : : : : : : : : : : : : : : : 1346.1 Work required to reduce norm of �rst-order residual by �ve ordersof magnitude (to nearest work unit). � = 10�; �rst-order upwindoperator with four-stage time-stepping. Description in the text. : : 1456.2 Work required to reduce norm of �rst-order residual by �ve orders ofmagnitude (to nearest work unit). Discretization with explicit resid-ual smoothing and corresponding optimized four-stage time-steppingcoe�cients. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1466.3 Work units required to reduce the norm of the �rst-order residualby �ve orders of magnitude. M = 0:35 test-case. Regular multigrid.64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1586.4 Work units required to reduce the norm of the �rst-order residualby �ve orders of magnitude. M = 0:35 test-case. Semi-coarsenedmultigrid. 64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : 159xv

6.5 Work units required to reduce kTEk1 to 10�2. M = 0:35 test-case. Second-order spatial operator (� = 0). Defect-correction cyclesused together with nested iteration for initial guess. Semi-coarsenedmultigrid. 64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : 1596.6 Work units required to reduce the norm of the �rst-order residualby �ve orders of magnitude. M = 0:85 test-case. Regular multigrid.64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1616.7 Work units required to reduce the norm of the �rst-order residualby �ve orders of magnitude. M = 0:85 test-case. Semi-coarsenedmultigrid. 64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : 1626.8 Work units required to reduce to reduce kTEk1 to 5 � 10�2. � = 0spatial discretization. Defect-correction cycles used together withnested iteration for initial guess. M = 0:85 test-case. Semi-coarsenedmultigrid. 64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : 1626.9 Work units required to reduce the norm of the �rst-order residualby �ve orders of magnitude. M = 1:4 test-case. Regular multigrid.64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1646.10 Work units required to reduce the norm of the �rst-order residualby �ve orders of magnitude. M = 1:4 test-case. Semi-coarsenedmultigrid. 64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : 1656.11 Work units required to reduce to reduce kTEk1 to 10�2. � = 0spatial discretization. Defect-correction cycles used together withnested iteration for initial guess. M = 1:4 test-case. Semi-coarsenedmultigrid. 64 � 32 grid. : : : : : : : : : : : : : : : : : : : : : : : : : 1656.12 Comparison of �rst-order convergence rates for ow past a semi-circular bump in a channel, 64x32 grid. Work units required toreduce the norm of the residual by �ve orders of magnitude. (Lo-cal TS � local time-stepping, Matrix TS � local preconditioning).Regular multigrid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1676.13 Comparison of �rst-order convergence rates for ow past a semi-circular bump in a channel, 64x32 grid. Work units required toreduce the norm of the residual by �ve orders of magnitude. Semi-coarsened multigrid. : : : : : : : : : : : : : : : : : : : : : : : : : : : 167xvi

6.14 Comparison of second-order (� = 0) convergence rates for ow pasta semi-circular bump in a channel, 64x32 grid. Work units requiredto reduce kTEk1 to 10�2 for M = 0:35 and M = 1:4 and to 5�10�2for M = 0:85. Nested iteration with 5 defect-correction sweeps oneach coarse grid level was used initially to improve robustness formultigrid solutions. Semi-coarsened multigrid. : : : : : : : : : : : : 1676.15 Comparison of �rst-order and second-order convergence rates for ow past a semi-circular bump on a wall, 64x32 grid, M = 0:85.Work units required to reduce the norm of the residual by �ve or-ders of magnitude (for �rst order) and kTEk1 to 5 � 10�2 (secondorder). Four-stage time-stepping schemes were used for all runs.Four grid-levels were used for the multigrid runs. (Local TS � localtime-stepping, Matrix TS � local preconditioning). Semi-coarsenedmultigrid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 169

xvii

CHAPTER IINTRODUCTIONMany aerodynamic problems of interest require the computation of a steady-state solution induced by some speci�c geometry. Important considerations whencomputing such ows are: the capability to treat complex geometries; the properrepresentation of shock waves, shear layers and contact discontinuities; and achiev-ing a high order of accuracy in the smooth parts of the ow. It is also importantto obtain these solutions with computational e�ciency and robustness. The mainobjective of this work is to contribute to the development of fast and e�cient nu-merical methods for the computation of steady solutions to the Euler equations of uid ow; in addition the extension to the Navier-Stokes equations is considered.The technique of marching in time is routinely used to solve steady-state problemsin computational uid dynamics. The presence of a time-derivative or pseudo-timederivative in the governing partial-di�erential equations allows the solution of aninitial/boundary-value problem instead of a boundary-value problem. The unsteadyEuler equations form a hyperbolic system in space-time, allowing for the use oftechniques derived for hyperbolic partial-di�erential equations. The time-derivativegoes to zero in the steady state, yielding the original time-independent system ofdi�erential equations. 1

2Numerical methods for time-integration can be broadly classi�ed into two cat-egories: explicit and implicit methods. Explicit marching schemes are becomingincreasingly popular because of the advantages they have over implicit formulations.Explicit schemes require less storage, are easier to implement on vector and parallelarchitectures and naturally allow for local grid re�nement or formulations on fullyunstructured grids.Explicit methods require more iterations than implicit methods, but each iter-ation is relatively cheap, both in terms of cpu as well as memory usage. One ofthe factors limiting the convergence speed of explicit methods is the time-step size,which is restricted by the Courant-Fredrichs-Levy (CFL) stability condition. Therehave been di�erent techniques proposed to accelerate the convergence of explicitschemes, including local time-stepping, residual smoothing [24], multigrid methods[7] and, most recently, local preconditioning [63]. Multigrid relaxation potentially isthe most powerful of these. For linear elliptic problems, multigrid achieves the ulti-mate goal of convergence in O(N) operations, where N is the number of unknowns;for mixed elliptic/hyperbolic problems with strong nonlinearity, such as generated bythe Euler equations, this goal is still far away. Local preconditioning addresses theissue of sti�ness in the governing equations by reducing the characteristic conditionnumber i.e., the ratio of the largest to the smallest characteristic speed, to the lowestpossible value over the entire Mach number range [33].It is the aim of this thesis to show that the inclusion of local preconditioning inmultigrid relaxation provides speci�c bene�ts (discussed below) that make multigridrelaxation a more e�cient and more robust method of obtaining steady-state solu-tions to the Euler equations. It should be mentioned that local preconditioning itselfis used with a local time-step. Furthermore, explicit residual-smoothing is shown to

3improve convergence rates further.The most popular explicit methods for computing steady solutions to the Eulerequations are the multi-stage methods pioneered by Rizzi [47] and Jameson el al.[25, 23]. These multi-stage methods must feature good high-frequency damping inorder to be suited for use with multigrid marching. Multi-stage methods do o�erthe exibility to be designed so as to achieve the desired smoothing properties for agiven spatial operator. Until recently, however, the design of optimally smoothingmulti-stage schemes was based entirely on the scalar one-dimensional [23, 66] ortwo-dimensional [10] convection equation.Tai's [66] procedure for optimizing the high-frequency damping in a one-dimensionalconvection scheme takes place in the complex plane, where the zeros of the multi-stage ampli�cation factor are designed to fall along the locus the Fourier transform -the \Fourier footprint" - of the discrete spatial operator. This can only be achievedfor one speci�c value of the time-step; �nding this value is part of the design process.For the two-dimensional Euler equations two complications arise. In the �rstplace, the Fourier footprint is no longer a one-dimensional curve, but a two-dimensionaldomain, making the optimization process more elaborate. In the second place, therenow are several kinds of physical signals propagating in all possible directions atdi�erent speeds; these are more or less accurately represented by the discrete opera-tor and produce di�erent concentrations of eigenvalues in its Fourier footprint. Thelack of clustering of high-frequency eigenvalues in the footprint makes it impossibleto obtain multi-stage coe�cients that produce good high-frequency damping of allhigh-frequency modes at all ow angles and Mach numbers.Local preconditioning as developed by Van Leer et al. [63] has the e�ect of equal-izing the characteristic (wave) speeds admitted by the Euler equations for the entire

4range of Mach numbers. One e�ect of local preconditioning on discretizations ofthe spatial Euler operator is a strong concentration away from the origin of the pat-tern of eigenvalues in the complex plane. It then again becomes possible to developmulti-stage coe�cients that boast low values of the amplication factor in the high-frequency concentrations of the footprints, thus providing systematic high frequencydamping for a particular discrete operator. This idea was �rst described by Van Leeret al. in [64].We have developed an optimization procedure to obtain sequences of time-stepvalues de�ning multi-stage schemes with superior smoothing properties, for dis-cretizations of the full Euler or Navier-Stokes spatial operator, provided these in-corporate local preconditioning. With the use of semi-coarsening rather than full-coarsening in the multigrid process [40], the high-frequency domain over which themulti-stage scheme must be a good damper of errors shrinks to \high-high" combina-tions only. This makes the multi-stage schemes largely independent of ow conditionssuch as Mach number and ow-angle.The resulting multi-stage schemes are not only preferable as solvers in a multigridstrategy, but are also superior single-grid schemes, in comparison to other explicitmarching schemes. The preconditioning itself already accelerates the convergenceto a steady solution (this bene�t persists in multigrid relaxation), and the high-frequency damping provides robustness [34, 35, 36]. This is a substantial improve-ment over the standard block-Jacobi preconditioning recommended for multigrid useby Mulder [40] and Allmaras [2], which yields equally good high-frequency dampingbut no equalization of characteristic speeds.The spatial-discretization techniques that we have selected in order to achievehigh resolution, e�ciency and robustness are upwind-based discretizations based on

5kappa-interpolation [61] together with Roe's approximate Riemann solver [48]. Thesetwo techniques are discussed in Chapter II.In this work, we restrict ourselves to the Euler and Navier-Stokes equations intwo dimensions. Furthermore, to avoid grid-generation issues, we restrict ourselves tostructured meshes. However, all techniques used can be extended in a straightforwardmanner to the three-dimensional Euler and Navier-Stokes equations, as explained inChapter V. The extension to unstructured meshes is also considered in Chapter V.Chapter II provides an introduction to multigrid methods and includes a descrip-tion of the spatial and temporal discretizations that we make use of in our multigridformulation. Chapter III describes aspects of local preconditioning, including theequations for the Fourier footprints that are used in deriving optimal multi-stageschemes.Chapter IV describes the formulation of the optimization problem and presentsoptimal schemes based on the solution to this problem. The ideas presented hereare extended in Chapter V; in particular we discuss the application to Navier-Stokesoperators, optimization formulations incorporating explicit residual-smoothing andthe extension to unstructured meshes and three dimensions.The analysis is illustrated with numerical experiments presented in Chapter VI.The results of analysis and experiments are summarized in Chapter VII, which alsoo�ers recommendations for future work.

CHAPTER IITHE MULTIGRID METHODThe amount of computational work should be proportional to the amountof real physical changes in the computed system. Stalling numerical pro-cesses must be wrong. A. Brandt.This statement by A. Brandt in [7] de�nes the motivating principle behind themultigrid method. Common examples of stalling in numerical processes are found inthe usual iterative processes for solving the algebraic equations arising from discretiz-ing partial-di�erential equations such as those governing boundary-value problems,in which the error changes little from one iteration to the next. Another exam-ple arises in the solution of time-dependent problems with time-steps much smallerthan the real scale of change in the solution, their value being dictated by stabilityrequirements [7].The multigrid method can help in such cases. The stalling in the problem is usu-ally related to some form of sti�ness that is present in the problem. This sti�nessis caused by the existence of solution components with di�erent, con icting scales.For example, in solving boundary value problems, smooth error components, whichcan be e�ciently approximated on coarse grids but which are slow to converge in�ne-grid processes, con ict with high-frequency error components that must be ap-6

7proximated on �ne grids. The multigrid method attempts to resolve such con ictsby employing several scales of discretization.Ideally, the computational e�ort required to solve a problem should be determinedonly by the amount of real physical information present, i.e., under quite generalconditions, the convergence rate of the iterative process should be independent ofproblem size. Such convergence rates have been achieved when solving problemsgoverned by linear elliptic partial-di�erential equations.True multigrid convergence rates are di�cult to obtain for solutions to problemsgoverned by hyperbolic partial-di�erential equations, even for linear cases.This chapter is meant to outline the multigrid method as utilized in this thesis.It is therefore rather restrictive in its description. The reader is referred to [7, 55, 28]as starting points for further study of the multigrid method and its application to uid dynamics.2.1 Elements of Linear MultigridConsider a linear system of the formLu = f (2:1)where L is a linear operator, and u and f are vectors. With an iterative scheme, ateach step an approximation v to the unknown solution u can be computed. De�nethe error in the solution as e = u� v: (2:2)Since the solution u is an unknown, this error e is also an unknown. Another measureof how well v approximates u, called the residual, may be de�ned:r = f � Lv: (2:3)

8By subtracting Equation (2.1) from (2.3) the residual equationLe = r (2:4)is obtained.Simple iterative methods, particularly explicit methods, can be very e�ective ineliminating the high-frequency or oscillatory components of the error, while leavingthe low-frequency or smooth components relatively unchanged. The immediate issueis whether these relaxation schemes can be incorporated in some way into a methodthat makes them act on all error components.One way to improve the relaxation process, at least in the early stages of conver-gence, is to use a good initial guess. This justi�es the use of the residual equation,since relaxation on the original equation (2.1) with an arbitrary initial guess v isequivalent to relaxing on the residual equation (2.4) with the speci�c initial guesse = 0.Assume now that a particular relaxation (iterative) scheme has been applied untilonly smooth error components remain. Figure 2.1 shows us what these modes wouldlook like on coarser grids. Consider a sinusoidal (Fourier) wave-mode of the formwhk;j = sin jk�N !where 0 � j � N and 1 � k � N � 1. The modes in the lower half of the spectrum,with wavenumbers in the range 1 � k < N=2, are the smooth modes, while modesin the upper half of the spectrum, with N=2 � k � N � 1 are the high-frequency oroscillatory modes. A wave with wavenumber k = 6 on a grid h, where h indicatesthe mesh width, with number of subintervals N = 16, is projected directly onto thecoarser grid 2h with number of subintervals equal to 8, i.e. N=2.

94h 0 1 2 3 4 5 6 7 80 2 4 10 12 14 166 8

0 1 2 3 4h2h

Figure 2.1: Discrete representation of a sinusoidal wave-mode (k = 6) on three dif-ferent grid levelsOn the coarse grid 2h the wave-mode maintains its wavenumber if 1 � k < N=2since whk;2j = sin 2jk�N ! = sin jk�N=2! = w2hk;j; 1 � k < N2 :However this wave-mode is no longer smooth on the coarse grid 2h. This grid canonly represent wave-modes with wavenumber k such that 1 � k � N=2 � 1, sincethere are only N=2 subintervals on this grid. Wave-modes with k � N=4 = 4 areoscillatory modes on this grid.It is worth noting that if this oscillatory wave-mode on 2h is projected onto aneven coarser grid 4h with 4 (i.e. N/4) subintervals, a phenomenon known as aliasingoccurs. The oscillatory mode is misrepresented as a relatively smooth mode on 4h.

10In general, the kth mode on a grid h becomes the (N�k)th mode on grid 2h whenk > N=2 and so on for other grid levels. The k = N=2 mode becomes the null-modeon 2h. The generation of smooth modes, which are harder to attack, is undesirable.Therefore, the relaxation scheme must be a good smoother of high-frequency error-modes to counteract the e�ect of aliasing.Leaving out aliasing, the point of note is that smooth wave-modes on a �ne gridlook less smooth on a coarse grid. This leads us to consider using coarser grids toexploit the strong smoothing of highly oscillatory wave-modes that is typical of mostgood relaxation schemes.2.1.1 Obtaining an initial guess with nested iterationWe elaborate on the idea of using coarse grids to obtain better initial guesses.The aim is to remove low-frequency (smooth) error-modes from the guess with afairly fast calculation. The procedure may be described as follows:1. Start with an initial guess on the coarsest grid.2. Relax on Lu = f on this grid so as to obtain an initial guess for the next (�ner)grid level.3. Repeat Step 2 until the �nest grid level is reached.This procedure will not damp out all the low frequency error-modes and furthercomputation is required to obtain a converged solution. However, it does provide agood initial guess to the solution, vh.

11Two-level multigrid for linear operatorsHaving relaxed on the �ne grid until convergence deteriorates, we relax on theresidual equation (2.4) on a coarser grid to obtain an approximation to the erroritself. We then return to the �ne grid to correct the approximation �rst obtainedthere.2.1.2 The Coarse-Grid Correction (CGC) schemeThe coarse-grid correction scheme may be written as follows:vh CGC(vh; fh)� Relax �1 times on Lhuh = fh on h with scheme P and initial guess vh to obtainan approximation to the solution vh P �1vh.� Compute the residual on the coarse grid r2h = I2hh (fh � Lhvh).� Solve the residual equation L2he2h = r2h exactly on 2h.� Interpolate e2h onto h and correct the �ne grid approximation: vh vh + Ih2he2h� Relax �2 times on Lhuh = fh on h with scheme P to obtain an approximation tothe solution vh P �2vh (smoothing step).Our description of multigrid schemes for linear operators closely follows the de-scription by Briggs [9]. We have also described multigrid schemes for nonlinearoperators using a similar format in later sections. I2hh and Ih2h are grid-transfer op-erators (restriction and prolongation operators respectively). These are designed totransfer discrete functions from one grid to another by averaging or interpolation (cf.[9]).

12Representing this scheme as a sequence of matrix operators we can writevh P �2 �P �1vh + Ih2h(L2h)�1I2hh (fh � LhP �1vh)� : (2:5)The exact solution uh is unchanged by the CGC scheme. Therefore, we may write,uh = P �2 �P �1uh + Ih2h(L2h)�1I2hh (fh � LhP �1uh)� : (2:6)Subtracting Equation (2.5) from (2.6) giveseh P �2 �I � Ih2h(L2h)�1I2hh Lh�P �1eh � Geh: (2:7)2.1.3 Two-level analysis in Fourier spaceThe most popular way to estimate multigrid convergence factors is by two-levelanalysis in Fourier space. Mulder has analyzed the two-dimensional linear-convectionoperator with periodic boundary conditions (cf. [42]). (See [56] for a description of theanalysis for the one-dimensional linear convection equation with periodic boundaryconditions).The multigrid convergence factor (with two-level analysis) is given by�� = max0�j�ij�� �(G) (2:8)whereG can be written as P �2KP �1 . The �i; i = 1; : : : ; d are the spatial frequencies inFourier space, equivalent to k�=N used earlier. d is the number of spatial dimensions)and �(:) is the spectral radius. K is the coarse-grid correction operator (CGC), givenby K = I � Ih2h(L2h)�1I2hh Lh:The coarse-grid correction operator should remove low-frequency iteration errors,and is used in combination with a smoother that removes high-frequency errors.

13Since the smoother is usually ine�cient for low frequencies, the convergence of low-frequency errors depends almost entirely on K. For the lowest frequencies (�i closeto zero), the restriction and prolongation operators have practically no e�ect, so thatK ' I � (L2h)�1Lh: (2:9)If the exact operator vanishes (Lu = 0), the discrete operator Lh equals thetruncation error � h. For these low-frequency waves considered above, then K '1� � h=� 2h. This implies that the worst case convergence rate for a scheme of orderp is at best 1 � 2�p, since � h=� 2h ' hp=(2h)p for such a scheme. This estimateis pessimistic at best, yielding a convergence rate of 7=8 for a third-order scheme.Mulder's analysis indicates that the actual two-grid convergence rate can be evenworse than this estimate. For example, if �rst-order restriction (volume-averaging)and prolongation (piecewise-constant interpolation) operators are chosen, the coarse-grid correction operator for the second- and third-order schemes are unstable (cf.[42]).The above implies that one cannot design a multigrid scheme with a uniformlygood convergence rate for a spatial discretization based on higher-order upwind dif-ferencing. However, as Tai [56] indicates, numerical experiments with a convectionoperator for simple problems not involving the propagation of a discontinuity haveshown that it is possible to make use of a higher-order scheme on all grids and con-verge quickly to a solution. This, however, is not necessarily the case for problemsinvolving discontinuities. Higher-order multigrid implementations will be discussedin more detail in the section on the defect-correction technique (Section 2.4).

142h(b)(a)(c)h4h8h2h

h4h8h2hh4h8h

Figure 2.2: Schedule of grids for (a) V-cycle, (b) W-cycle and (c) Sawtooth cycle, allon four levels2.1.4 The V-Cycle and the �-CycleThe following question arises in the solution of the residual equation on the coarsegrid in the CGC scheme described above: What is the best way to solve the coarsegrid problem L2he2h = r2h ?The coarse grid problem is not much di�erent from the original problem. Wecan therefore apply the CGC scheme to the residual equation on 2h, which meansmoving to 4h for the correction step. This process can be repeated on successivelycoarser grids until a direct solution of the residual equation is possible.The algorithm telescopes down to the coarsest grid, which can be a single interior

15grid point, and then works its way back to the �nest grid. Figure 2.2 (a) shows theschedule for the grids in the order in which they are visited. This cycle is called theV-cycle, based on the pattern in this �gure. The x-axis in the �gure can be taken tobe either work or time.The V-cycle has a compact recursive de�nition, which may be written as follows:vh MV h(vh; fh)� Relax �1 times on Lhuh = fh on h with scheme P and initial guess vh to obtainan approximation to the solution vh P �1vh.� If h 6= coarsest grid then� f2h I2hh (fh � Lhvh)� v2h 0� v2h MV 2h(v2h; f2h)� Correct vh vh + Ih2hv2h.� Relax �2 times on Lhuh = fh on h with scheme P to obtain an approximation tothe solution vh P �2vh.Note that we are solving the residual equation on coarser grids, though the no-tation may seem to indicate otherwise.The V-cycle is just one of a family of multigrid cycling schemes known as the�-cycle family. This family can be de�ned recursively as:vh M�h(vh; fh)� Relax �1 times on Lhuh = fh on h with scheme P and initial guess vh to obtain

16an approximation to the solution vh P �1vh.� If h 6= coarsest grid then� f2h I2hh (fh � Lhvh)� v2h 0� Update v2h M�2h(v2h; f2h) � times� Correct vh vh + Ih2hv2h.� Relax �2 times on Lhuh = fh on h with scheme P to obtain an approximation tothe solution vh P �2vh.� = 1 corresponds to the V-cycle and � = 2 corresponds to the W-cycle (Figure2.2(b)). These are the two schemes used in practice.These cycling schemes are also commonly used with nonlinear systems. Howeverthe details of their implementation di�er.2.2 Aspects of Nonlinear MultigridMost of the ideas described this far carry over to the solution of nonlinear prob-lems. One important exception is the coarse-grid correction scheme. The residualequation that is used in the coarse-grid correction scheme (2.4) is not valid for non-linear operators. It is therefore necessary to consider an alternate approach on thecoarse grid. The algorithm incorporating this alternate approach is commonly knownas the Full Approximation Storage (FAS) method [6].2.2.1 Moving from CGC to FASAs before, consider a linear problemLu = f : (2:10)

17The approximation to this problem on the �ne grid can be written asLhuh = fh: (2:11)The residual rh with an approximate solution vh to (2.11) is given byrh = fh � Lhvh: (2:12)In the CGC scheme, the coarse-grid unknown function is e2h, intended to approxi-mate the correction function eh on the �ne grid. Since it is not always possible toformulate and solve an equation for e2h (e.g. with nonlinear operators), FAS avoidssolving for e2h on the coarse grid, solving instead an approximation to the �ne-gridproblem: L2hu2h = f2h (2:13)where f2h = L2h(I2hh vh) + I2hh rh: (2:14)(I2hh and I2hh are both used here to indicate that the restriction operators for thesolution and the residual can be di�erent). In the analog to the CGC scheme, thisequation is solved exactly and the approximate solution on the �ne-grid is correctedas follows: vh vh + Ih2h �u2h � I2hh vh� : (2:15)Three points are worthy of note:� For linear problems, the FAS steps described here are equivalent to the stepsused in CGC.� Using vh Ih2hu2h instead of Equation (2.15) is not a good choice, as thiswould introduce the interpolation errors due to the full solution u2h instead ofthe interpolation errors due to the correction e2h � u2h � I2hh vh.

18� The same interpolation operator I2hh should be used in Equations (2.14) and(2.15). Any di�erence may dominate the calculated correction function andhinder convergence.2.2.2 FAS with nonlinear problemsAs mentioned above, CGC is not applicable to nonlinear problems directly asthe residual equation Lheh = rh is valid only for linear problems. FAS is directlyapplicable however.Consider the following nonlinear problem:Nhuh = fh (2:16)Assume that we obtain an approximation vh to the solution on the �ne grid. Theneh = uh � vh (2:17)de�nes the error in the approximate solution. Substituting Equation (2.17) into(2.16), we obtain Nh(vh + eh) = fh: (2:18)We may also write the residual asrh = fh �Nhvh (2:19)Combining Equations (2.18) and (2.19) we obtain the following equation for theresidual: Nh(vh + eh)�Nhvh = rh: (2:20)Transferring this equation to the coarse grid we obtain:N2h �I2hh vh + e2h��N2h �I2hh vh� = I2hh rh: (2:21)

19This may be written as N2hu2h = I2hh rh +N2h �I2hh vh� (2:22)which is the analog to the �ne-grid problem with right-hand side f2h = I2hh rh +N2h(I2hh vh). Also u2h = I2hh vh+ e2h. Equation (2.22) is solved for u2h as before andthe correction e2h = u2h � I2hh vh is prolongated back to the �ne grid. The �ne-gridapproximation is now updated.vh vh + Ih2h �u2h � I2hh vh� : (2:23)As before, this two-grid scheme can be extended to more than two grid levels.The nonlinear analog to the linear V and � cycles can now be formulated.2.2.3 Nonlinear versions of the V and � cyclesThe nonlinear version of the V-cycle incorporating FAS has a compact recursivede�nition that may be written as follows:vh NMV h(vh; fh)� Relax �1 times on Nhuh = fh on h with scheme P to obtain an approximation tothe solution vh.� If h 6= coarsest grid then� rh fh �Nhvh� f2h I2hh rh +N2h(I2hh vh)� v2h I2hh vh� v2h NMV 2h(v2h; f2h)� Correct vh vh + Ih2h �v2h � I2hh vh� .

20� Relax �2 times on Nhuh = fh on h with scheme P to improve the approximation tothe solution vh.The family of schemes that this cycle belongs to can be written as:vh NM�h(vh; fh)� Relax �1 times on Nhuh = fh on h with scheme P to obtain an approximation tothe solution vh.� If h 6= coarsest grid then� rh fh �Nhvh� f2h I2hh rh +N2h(I2hh vh)� v2h I2hh vh� Update v2h NM�2h(v2h; f2h) � times� Correct vh vh + Ih2h �v2h � I2hh vh� .� Relax �2 times on Nhuh = fh on h with scheme P to improve the approximation tothe solution vh.As before, � = 1 corresponds to the V-cycle and � = 2 corresponds to theW-cycle. These are the two schemes used in practice.It is possible to apply the linear multigrid schemes to the Newton linearizationof Nh around the current approximation vh. However, the nonlinear FAS versionsare usually preferable, since:� No global linearization is required. The only linearization that may be requiredis local linearization in the relaxation scheme.

21� Programming FAS is modular, since the FAS equations are exactly the sameas the original discrete equations, except with a di�erent right hand side.� The FAS multigrid rate of convergence is not coupled to the rate of convergenceof Newton iterations. It is mainly determined by the interior smoothing rate.� Newton methods require continuous di�erentiability of the scheme. This isnot generally satis�ed, in particular, not by Roe's upwind ux function [48].Continuously-di�erentiable upwind schemes do exist though [44, 60].Both the linear and nonlinear multigrid approaches have been used in multigridsolution of the Euler equations. Examples of the linear approach are [39, 26]. Exam-ples of the nonlinear approach are [21, 23]. Crucial to the success of both approachesis the need for high-frequency components to be e�ciently damped on the �ne gridlevel.The Sawtooth FAS cycleTaking � = 1 and �2 = 0 in routine NM�h() gives us a cycling scheme known asthe Sawtooth FAS scheme (Figure 2.2(c)). The Sawtooth FAS cycle does away withthe post-prolongation smoothing steps at each intermediate grid level. Jameson [23]has used such a formulation to good e�ect.2.2.4 Choice of restriction and prolongation operatorsFor each coarse grid cell the restricted solution is an area-weighted average of itsfour included �ne cells,v2kh I2khkh vkh = 1A2kh 4Xi=1 �vkhAkh�i (2:24)

22where k represents an intermediate �ne-grid level. A is the cell area of the corre-sponding cell. For k = 1 this reduces tov2h I2hh vh = 1A2h 4Xi=1 �vhAh�i (2:25)where, as before, h and 2h refer to the �ne and coarse grids respectively.We make use of bilinear interpolation for the prolongation operator Ih2h. Refer to[1, 56] for more information on restriction and prolongation operators.2.3 A Nonlinear Variant - Semi-Coarsened MultigridWe have assumed this far that there is a reasonable amount of coupling betweenneighboring nodes or cells in the discretization. This coupling can be exploited by asuitable relaxation scheme so as to remove oscillatory components of the error.However, the relaxation scheme may be unable to remove all the oscillatory com-ponents as required by the multigridmethod if the coupling between cells or nodes be-comes too small in one of the coordinate directions. In a multi-dimensional problem,the coupling can become locally one-dimensional. Consider the problem of comput-ing a steady solution to the (hyperbolic) linear convection equation ut+aux+buy = 0for b! 0, a 6= 0. In this example the di�erential operator becomes one-dimensionaland the solution will be independent of y unless some structure is imposed in the y-direction by boundary conditions. A proper discretization would re ect this by beingindependent from neighboring values in the y-direction. Consider now an error-modethat has an arbitrary structure in the x-direction and that is highly oscillatory inthe y-direction. This error-mode cannot be removed on the �ner grid either as aresult of the lack of coupling in the y-direction. Neither can this error-mode be re-moved on a coarser grid since the high-frequency oscillatory component cannot be

23represented appropriately on this grid. This is a well-known complication that ariseswhen modeling problems, even elliptic ones, that exhibit strong anisotropy.The problem of decoupling in one of the coordinate directions is termed strongalignment by Brandt [7]. In practical applications such as transonic ow about anairfoil using an O-grid, alignment only occurs in small regions. Alignment is fairlylikely to occur in subsonic channel ow discretized on a structured mesh. Cartesianunstructured meshes (cf. [72]) are likely to be subject to alignment to some extentas well. The issue is not of concern for fully unstructured meshes, such as thoseobtained by triangulating the domain.One remedy for the problem of alignment is to use global relaxation schemes suchas line-relaxation (Gauss-Seidel and its symmetric variants can only partly handlealignment and should be avoided as relaxation schemes in a regular multigrid methodfor problems exhibiting alignment). Mulder [41] indicates that for the Euler equa-tions in two space dimensions, alternating-direction damped line-Jacobi can providea uniformly good convergence rate. However, in three-dimensions, plane-relaxationwould be necessary, which is not an attractive option. Also, line-relaxation is gen-erally restricted to simple domains, whereas a multigrid method with a relaxationscheme that uses only local data can in principle be used on domains of arbitraryshape.Another way to solve the problem of alignment is by using semi-coarsening. Withsemi-coarsening, cells are combined in the direction of strongest coupling. Any oscil-latory component of the error in the direction of weak coupling will then be brought tothe coarser grid. In [40] Mulder has derived a �rst-order spatially-accurate methodthat is based on semi-coarsening in multiple directions simultaneously. A higher-order solver based on semi-coarsening is also described by Mulder in [42].

242.3.1 Mulder's semi-coarsened multigrid methodConsider semi-coarsening in two directions simultaneously. From one �nest gridof 8 � 8 points, two coarser grids, with 4� 8 and 8� 4 points are obtained. On thenext level, four grids are created and so on. In this way, the total number of grids isdoubled after each restriction and the total number of points in all grids on each levelremains constant. In three dimensions however, the number of points increases withprogressively coarser grids. For example, starting from an 8�8�8 grid, the numberof grids is tripled, with each grid carrying half the number of points, so that the totalnumber of points would increase by a factor of 3=2 on the next level. Proceedingto still coarser grids further increases storage and operations requirements. It istherefore necessary to prune some of the grids created on coarser levels in order toobtain a practical method based on semi-coarsening.The complexity can be reduced by choosing the sequence of grids in Figure 2.3.This method requires a modi�cation of the standard multigrid algorithm to accountfor the multiple parent and child grids that occur with this sequence. Mulder indi-cates that this method has O(N) cost per multigrid cycle if a V- or F-cycle is used.The reader is referred to [40] for his analysis of computational complexity.The description of the restriction and prolongation operators that follows assumesthat an FAS scheme is being used to solve the nonlinear problemNhuh = fh:Let the �nest grid have 2M1 points in the x-direction and 2M2 points in the y-direction,resulting in a total of 2M1+M2 points. The level number L corresponding to this gridis de�ned by L = 1. The set of all coarser grids, including this �ne grid, is de�nedby the integer pair (m1;m2), where 0 � m1 � M1 and 0 � m2 � M2 and each grid

25(7)

8,8 8,42,8 4,4 8,21,8 2,4 4,2 8,11,4 2,24,8 4,11,2 2,11,1(1)(2)(3)(4)(5)(6)Figure 2.3: Arrangement of �nest (8� 8) and coarser grids for problems with align-ment. The arrows indicate how the grids are linked by restriction (down-ward) and prolongation (upward).has dimensions 2m1 � 2m2 . The level number corresponding to this intermediate gridis given by L(m1;m2) = (M1 +M2) � (m1 + m2) + 1. This de�nition of the levelnumber allows us to keep our description consistent with the notation used in earliersections and di�ers from Mulder's treatment.Restriction and prolongation operatorsWe will �rst describe the restriction operator. As before, a grid is uniquelyde�ned by (m1;m2). We denote the current solution on this grid by v(m1;m2), andthe corresponding residual byr(m1;m2) = f (m1;m2) �N (m1;m2) �v(m1;m2)� :We denote the restriction operator for the residual by Im1;m2m1+1;m2 and the restrictionoperator for the solution by Im1;m2m1+1;m2 for semi-coarsening in the x-direction. Similarly

26for semi-coarsening in the y-direction we use Im1;m2m1;m2+1 for the residual-restriction op-erator and Im1;m2m1;m2+1 for the solution-restriction operator. An algorithmic descriptionof the operations involved in restriction from level l � 1 to l is given by:for all (m1;m2) with L(m1;m2) = l; 0 � m1 �M1; 0 � m2 �M2 doif there are links to (m1 + 1;m2) and (m1;m2 + 1) thenv(m1;m2) 12 hIm1;m2m1+1;m2v(m1+1;m2) + Im1;m2m1;m2+1v(m1;m2+1)if (m1;m2) 12 hIm1;m2m1+1;m2r(m1+1;m2) + Im1;m2m1;m2+1r(m1;m2+1)i+N (m1;m2) �v(m1;m2)�else if there is only one link to (m1 + 1;m2) thenv(m1;m2) Im1;m2m1+1;m2v(m1+1;m2)f (m1;m2) Im1;m2m1+1;m2r(m1+1;m2) +N (m1;m2) �v(m1;m2)�else if there is only one link to (m1;m2 + 1) thenv(m1;m2) Im1;m2m1;m2+1v(m1;m2+1)f (m1;m2) Im1;m2m1;m2+1r(m1;m2+1) +N (m1;m2) �v(m1;m2)�end ifend doThe resulting coarse-grid problem becomesN (m1;m2) �u(m1;m2)� = f (m1;m2): (2:26)This equation could be solved exactly, or by a multigrid approach, as before.The basic di�erence between this restriction operator and operators for the standardmultigrid approach is the occurence of two links to �ner grids. We use equal weightingof data from the two grids, as in Mulder's original reference. This gives no speci�cbias to any of the coordinate directions.

27The prolongation operator brings corrections from one or more coarser grids to thecurrent grid. An algorithmic description of the operations involved in prolongationfrom level l + 1 to l is given by:for all (m1;m2) with L(m1;m2) = l; 0 � m1 �M1; 0 � m2 �M2 doif there is a link to (m1 � 1;m2) thenv(m1;m2) v(m1;m2) + Im1;m2m1�1;m2 hv(m1�1;m2) � Im1�1;m2m1;m2 v(m1;m2)iend ifif there is a link to (m1;m2 � 1) thenv(m1;m2) v(m1;m2) + Im1;m2m1;m2�1 hv(m1;m2�1) � Im1;m2�1m1;m2 v(m1;m2)iend ifend dowhere Im1;m2m1�1;m2 and Im1;m2m1;m2�1 are the prolongation operators associated with semi-coarsening. (Im1�1;m2m1;m2 and Im1;m2�1m1;m2 are the solution-restriction operators introducedearlier).The fundamental di�erence between this prolongation operator and those of stan-dard multigrid is that the corrections are always computed with respect to the re-striction of the most recent solution rather than the solution at the beginning ofthe multigrid cycle. Choosing the x-direction �rst in the algorithmic loop resultsin an asymmetry within the algorithm with respect to coordinate directions. Thisasymmetry is reduced by alternating the order of prolongation in x and y each timethe prolongation step is carried out.The approach described above takes care not to introduce any bias with respectto coordinate directions. In some cases however, such a bias may be bene�cial.For example, biasing prolongation in the direction of convection for a convection-

28dominated ow would be expected to improve convergence. (Koren [29] describesone such algorithm that is used to solve the Euler equations in conjunction with thestandard multigrid approach). Naik and Van Rosendale [43] have applied a variantof Mulder's algorithm to anisotropic elliptic problems. Their variant combines thecontributions from the multiple coarse grids via a local switch that is based on thestrength of the discrete operator in each direction. This improves the convergencerate of the method.Extending the method to higher orderAs with the standard multigrid approach, convergence rates are poor for higher-order discretizations. Two-grid analysis with the linearized Euler equations [42]shows that it is di�cult to obtain uniformly good convergence rates for a higher-order scheme, because of waves perpendicular to stream lines. Though the defect-correction technique, described below, su�ers from the same problem, it may be usedto provide convergence to a point where the residual is of the order of truncationerror in relatively few multigrid cycles.2.4 The Defect-Correction TechniqueIt is harder to obtain good smoothing properties for relaxation schemes incor-porating higher-order spatial-discretizations than it is for �rst-order discretizations.This implies that multigrid convergence is likely to be much worse for such schemesthan it is for a �rst-order scheme.One way to improve damping of high-frequency error components, and henceimprove multigrid convergence, is by explicitly adding some arti�cial di�usion to thescheme. However, this reduces accuracy. Furthermore, adding the right amount of

29di�usion is likely to be a matter of trial and error.A second possibility is to solve the higher-order problem in an indirect way -by making use of the good multigrid convergence properties of the multigrid solverincorporating the �rst-order scheme. This may be achieved by the defect-correctiontechnique. The defect-correction multigrid technique is a special case of the conceptof defect corrections [3]. The reader is referred elsewhere for mathematical detailssuch as convergence bounds (cf. [55]).The basic idea behind a defect-correction step is that, given a solver for the lower-order discrete systemNhLuh = fh, a correction may added to fh on the currently-�nestgrid so as to raise the approximation order. The correction term is given byQ = NhLvh �NhHvh (2:27)where NhH is the higher-order operator and vh is the current approximate solution.(A similar correction is introduced to the discrete boundary conditions as well).The purpose of defect correction is to correct low-frequency components. Higher-order approximations, such as NhH , are better than lower-order approximations likeNhL in representing these low-frequency components. Recognizing this and the factthat in multigrid processes low frequencies are converged via the coarse-grid cor-rections, we see that the main e�ect of the defect corrections can be obtained byapplying them at the stage of transferring residuals to the coarser grids.Another interesting property of the defect-correction method is that the higher-order operator need not be stable (cf. [7, 55]).2.5 Full Multigrid (FMG)It is possible to extend the idea of nested iteration (Section 2.1.1) by obtainingthe initial guess on each intermediate grid by means of a multigrid cycle over the

303h4h2hhFigure 2.4: Full Multigrid cycle incorporating a single (�3 = 1) Sawtooth-FAS cycleon each intermediate grid levelcoarser grids. This approach is known as the Full Multigrid (FMG) algorithm.FMG provides a good initial guess for the �ne-grid solution. The solution may beconverged further with additional �ne-grid multigrid cycles. Brandt [7] indicates thatone FMG cycle is su�cient to converge the solution to within the level of truncationerror on the �ne grid. This is di�cult to obtain in practice, particularly for thecomplex nonlinear systems that we are interested in solving. However, as indicatedbefore, FMG provides an excellent �rst guess and may therefore be used to reducethe work required to obtain a converged solution.An algorithmic description of the steps involved is given below:vh FMGh(vh; fh)� If h 6= coarsest grid then� v2h FMG2h(v2h; f2h)� Interpolate vh ~Ih2hv2h� Update vh NM�h(vh; fh) �3 timesIn practice, this recursive form is not the best way to code up the scheme. It

31is more e�cient to code up the scheme as a loop starting on the coarsest grid andending up on the �nest grid.The FMG interpolation operator ~Ih2h is not necessarily the same as the correction-interpolation (prolongation) operator Ih2h. Brandt [7] indicates that, in general, theorder of ~Ih2h should be higher that the order of Ih2h, since the �rst approximation issmoother than the corrections. Our experiments indicate that using bilinear interpo-lation for both operators is su�cient to provide a good initial guess to the �ne-gridsolution.It is also possible to apply defect-correction cycles as the update steps in thealgorithm described above. This is the approach we utilize in obtaining good initialsolutions for our higher-order solver. We make use of Sawtooth-FAS cycles in all ournumerical experiments.2.6 Elements of the Update SchemeThe Euler equations of gasdynamics describe the ow of an inviscid noncon-ducting compressible gas; for ow in two space dimensions they may be written asfollows: @U@t + @F@x + @G@y = 0 (2:28)where U = ( �; �u; �v; �E )Tand F = 0BBBBBBBBBBB@ �u�u2 + p�uv�uH 1CCCCCCCCCCCA ; G = 0BBBBBBBBBBB@ �v�uv�v2 + p�vH 1CCCCCCCCCCCA;

32here �, u, v and p are the primitive variables and E and H denote speci�c totalenergy and enthalpy. The independent and dependent variables have been nondi-mensionalized by the following procedure:x = x�L ; y = y�L ; t = t�L=U1u = u�U1 ; v = v�U1 ; p = p��1U21 (2:29)and e = e�U21where L is a characteristic length scale, U1 a reference velocity, and e denotes speci�cinternal energy. For a perfect gas we have pressurep = ( � 1)�e= ( � 1) �E � �2 �u2 + v2�� (2.30)and local speed of sound c = s p� : (2:31)This is the conservation-law form of the Euler equations. Use of the conservation-lawform allows shock waves to be captured as weak solutions to the governing equationsand avoids the di�culties inherent in the application of shock-�tting techniques toarbitrary ows.2.6.1 Spatial discretizationWe make use of an upwind-di�erence scheme for our spatial discretization. Up-wind di�erencing models the wave-propagation nature of the equations in the sensethat information at each grid point is obtained from directions dictated by character-istic theory. Upwind methods have the advantage of being naturally dissipative; sep-arate spatial dissipation terms, such as are generally required in a central-di�erence

33method to overcome oscillations or instabilities in regions with strongly varying gra-dients, need not explicitly be added.Van Leer et al. [67] have shown that, for an accurate representation of boundarylayers, the ux formula must include information about all di�erent waves (nonlinearand linear) by which neighboring cells interact, as in Roe's ux-di�erence splitting[48]. Since we are ultimately interested in Navier-Stokes applications, we chose Roe's ux-di�erence splitting as our particular method of providing upwind bias. Roe's ux formula is one of a class of formulas that obtain the interface ux from anapproximate solution to the Riemann problem of gasdynamics.The �nite-volume form of the Euler equations that is used may be written as @Uij@t !n = � 1Aij( mXl=1 (F�y �G�x)l)ij (2:32)where Uij represents the vector of cell-averaged conserved quantities for cell (i, j).Aij is the area of cell (i, j). The index l denotes the sequence of cell faces on the cellboundary and (�x)l, (�y)l are the changes of x and y along the lth cell face goingin the counter-clockwise direction. Fl and Gl are the cell-face averaged values of the ux vectors in the Cartesian reference frame.The cell-face length is given by(�s)l = q(�x)2l + (�y)2l (2:33)and we may write the outward velocity normal to the cell face asql = (u�y � v�x)l=(�s)l (2:34)and the velocity parallel to the cell face asrl = (u�x+ v�y)l=(�s)l : (2:35)

34The outward ux normal to the cell face l based on cell-centered values is(F�y �G�x)l = 0BBBBBBBBBBB@ �q�qu+ p�y�s�qv � p�x�s�qH 1CCCCCCCCCCCA�s � �l�s : (2:36)Roe's Approximate Riemann Solver in two dimensionsUsing the usual notation with subscripts L and R to represent the left and rightstates across the interface, Roe's ux function is given by�(UL;UR) = 12 f�(UL) +�(UR)g � 12Rj�j�V (2:37)where �V = 0BBBBBBBBBBB@ (�p � �c�q)=(2c2)��r=c����p=c2(�p � �c�q)=(2c2) 1CCCCCCCCCCCA (2:38)and R = 2666666666664 1 0 1 1u� c�y�s c�x�s u u+ c�y�sv + c�x�s c�y�s v v � c�x�sH � qc rc 12(u2 + v2) H + qc 3777777777775 : (2:39)The hats ( ) used above refer to Roe averages. The Roe average of a quantity,speci�cally, of 1=�, u, v or H, is computed by weighting the quantity with p�, forinstance: u = uLp�L + uRp�Rp�L +p�R ; (2:40)

35all other \hatted" quantities are derived from these by their functional relationships.Also, the � operator in Equation (2.38) refers to the change between the left andright states, i.e. �(:) = (:)R � (:)L : (2:41)The matrix j�j is the diagonal matrix of absolute characteristic speeds jakj.j�j = 2666666666664 jq � cj 0 0 00 jqj 0 00 0 jqj 00 0 0 jq + cj 3777777777775 : (2:42)Entropy �x for Roe's SchemeThe scheme described above permits the existence of expansion shocks. This cana�ect the convergence rate of the solution and may even result in incorrect (non-physical) solutions. The absolute characteristic speeds may be modi�ed to satisfythe entropy condition and hence avoid such situations [19, 62]. For jakj�; k = 1; 4these speeds may be computed as follows:�a = aR � aL (2.43)�a� = 4 max (�a; 0) (2.44)jaj� = 8>><>>: jaj if jaj > 12�a�a2�a� + 14�a� if jaj � 12�a� (2.45)This eliminates expansion shocks, and ensures a smooth transition from subsonic tosupersonic ow.This form of the ux function requires some modi�cation if it is to be used withlocal preconditioning. This modi�cation is described in more detail in the nextchapter (Section 3.6).

36Higher-order spatial discretizationThe scheme described above is formally only of �rst-order accuracy. In order toraise the order of accuracy of the upwind di�erencing, all one needs to do is to raisethe order of accuracy of the initial-value interpolation that yields the zone-boundarydata. We make use of Van Leer's family of �-schemes [61], which obtain interfacevalues as follows:qLi+ 12 = �qi + 14s ��i�i �q;�i+i �q� h(1 � �s)�i�i + (1 + �s)�i+i )i �q (2:46)qRi� 12 = �qi � 14s ��i�i �q;�i+i �q� h(1� �s)�i+i + (1 + �s)�i�i )i �q (2:47)where �i+i �q = �qi+1 � �qi (2:48)and �i�i �q = �qi � �qi�1 : (2:49)The function s(���q;�+�q) is a limiter that prevents numerical oscillations. Wemake use of a smooth limiter due to Van Albada [59]:s ����q;�+�q� = 2(���q�+�q + �2a)(���q)2 + (�+�q)2 + 2�2a (2:50)With the proper use of a limiter, steady discontinuities are spread by the �-schemes over only 2 cells without oscillations. These schemes are hence known ashigh-resolution schemes.If the limiter is not used (s(���q;�+�q) = 1), the standard second-order scheme(Fromm's scheme [15]) is obtained for � = 0. The choice � = �1 provides a fullyone-sided upwind scheme [5]. For � = 13, third-order accuracy is obtained in avolume-averaged sense (only in one-dimension though).

37If the grid is not close to Cartesian, or if cell-sizes vary strongly from one cellto another, the one-dimensional interpolation (2.46) might need to be corrected forstretching and curvature. We make the assumption that the grid is locally almostCartesian, without signi�cant stretching from cell to cell. Note that the aspect ratioof the cells can be far away from unity.2.6.2 Boundary conditionsThe upwind scheme described above solves the Riemann problem at every in-ternal cell-interface in order to obtain an estimate of the interface ux. It is veryconvenient to extend this approach to the boundaries as well. Since the ux-formulaautomatically selects the proper information, one may add in an extra cell across theboundary in which the full state is prescribed at any time, even for subsonic in owand out ow. This overspeci�cation works well with any reasonable assumption aboutexterior states, but is not completely legitimate. The redundant information fromthe exterior (ghost) cell could in uence the Roe matrix and hence a�ect the solutionnear the boundary. This is not a problem if the in ow and out ow conditions aresu�ciently close to freestream, as in our test cases. We prescribe in ow, out ow andfar-�eld boundary conditions in this manner.Wall boundary conditions may prescribed in a similar manner. One may imaginethat across the wall, the gas meets its \mirror image". For inviscid ow (perfectslip) this would amount to setting the velocity component normal to the wall, vn, inthe ghost cell, equal and opposite to vn in the cell next to the wall. The values ofdensity and pressure remain unchanged. The ghost-cell wall-boundary procedure isof �rst-order accuracy because it neglects wall curvature. A correction to the ghost-cell state value is given by Dadone and Grossman [11]. The ghost-cell formulation

38uuBFigure 2.5: Obtaining the boundary-interface ux with the ghost-cell approach: the ux is obtained by solving the associated Riemann problem with thestates in the ghost cell and boundary cell as initial states.is also more dissipative than extrapolating the pressure at the wall from the interiorsolution.However, the ghost-cell approach may be easily applied to the preconditionedEuler equations as well. Since the preconditioned equations admit the same steadywaves as the un-preconditioned equations, all we need is a boundary procedure thatis stable and that guarantees the same steady-state solutions with preconditioningas without preconditioning. The ghost-cell approach turns out to be in this category(cf. [33]).The approaches described above lead to discretizations that are �rst-order accu-rate at the boundary. We have used these �rst-order accurate boundary conditionstogether with the high-resolution schemes described above. There is a well knownrule of thumb that a local discretization that is one order lower in accuracy than theoverall discretization does not change the global accuracy of the discretization [18],so this approach is expected to lead to accurate solutions.

392.6.3 Temporal discretizationThe most popular explicit methods for marching toward steady solutions of theEuler and Navier-Stokes equations are the multi-stage methods pioneered by Rizzi[47] and Jameson et al. [25, 23]. When updating the solution ofUt = Res(U) (2:51)from time level tn to tn+1 = tn +�t, an m-stage method takes the formU(0) = Un; (2.52)U(k) = U(0) +�t(k)Res �U(k�1)� ; k = 1; ::;m; (2.53)Un+1 = U(m) (2.54)with �t = �t(m): The m coe�cients �t(k) o�er the exibility needed to designmarching schemes suited for multigrid application.In a multigrid procedure, one special task of the marching scheme is to removehigh-frequency components of the error while marching. To make a marching schemea good multigrid smoother, the temporal and spatial discretization must be matchedto each other. Since the spatial discretization dictates the �nal accuracy of thesolution, the most natural way to achieve this is to select a spatial discretizationand to then design the time discretization in such a way that oscillatory waves aree�ectively damped.A major portion of this thesis (Chapter IV) is dedicated to obtaining the coe�-cients of multi-stage schemes that systematically damp high-frequency error-modesfor a system of equations, in particular the Euler equations. For a system of equa-tions such as the Euler equations, local preconditioning, i.e. multiplying the residualin Equation (2.51) by a matrix P(U), plays an important role in ensuring gooddamping. In general,

40� Preconditioning removes the sti�ness due to the spread of characteristic wavespeeds, improving single-grid convergence.� The e�ect of a local preconditioning matrix on discretizations of the spatialEuler operator is a strong concentration away from the origin of the pattern ofeigenvalues in the complex plane.The latter e�ect makes it possible to design multi-stage schemes that systemati-cally damp most high-frequency waves admitted by the particular discrete operator.Local preconditioning of the Euler equations is discussed in more detail in thenext chapter.2.7 Implications for Parallel ProcessingThe two issues of concern on a distributed-memory machine (including machineswith a single address-space such as the KSR) are load-balancing among processorsand minimizing the communication between processors.The update scheme described above is local in nature. The spatial discretizationmakes use of Van Leer's family of � schemes. These schemes are fairly compact,utilizing, in each step, information only from neighboring cells. Multi-stage temporalupdates are also local in nature. The state in each cell may be advanced throughone stage in time independently of other cells in the mesh. The update schemetherefore allows for �ne-grained parallelism, with possibly a few cells being mappedto each processor in a massively parallel setup. Load balancing becomes a fairlytrivial problem if the method is parallelized at the level of the update scheme. Theproblem of partitioning such a mesh (in particular - the more complex problem ofpartitioning an unstructured mesh) so as to achieve load balancing while reducing thecommunication overhead between processors has been well-researched (cf. [52, 69]).

41The increased robustness associated with Mulder's semi-coarsening method isobtained at a price. This method is more expensive than regular multigrid (especiallyon a single processor) because of the large number of coarse grids involved. Oneimportant property though of Mulder's semi-coarsening algorithm is that multiplecoarse grids on the same level are independent of each other. This improves thealgorithm's performance on a parallel processor. Load balancing becomes a morecomplex problem in this case. Overman and Van Rosendale [45] describe two possiblestrategies for load balancing. The communication-overhead incurred is considered aswell in this reference.

CHAPTER IIILOCAL PRECONDITIONINGThe hyperbolic system formed by the unsteady Euler equations becomes sti�when the speeds of the waves described by these equations become vastly di�erent,i.e., the ratio of the largest to the smallest characteristic speed, the characteristiccondition number, becomes large. This happens in almost incompressible (low Mach-number) ow, stagnation ow and transonic ow. In these situations, the convergencerate and even the accuracy of the numerical computations can deteriorate [71].Local preconditioning (preconditioning the residual by a local matrix) takes awaythe sti�ness due to the variation among the local characteristic speeds. Time accu-racy is lost as a result, but convergence to the steady state, especially for subsonicand transonic solutions, may be accelerated dramatically [63].Another e�ect of local preconditioning is to concentrate patterns of eigenvaluesof the discrete spatial operator in the complex plane, away from the origin. Thismakes it possible to design multi-stage schemes that systematically damp most high-frequency waves admitted by this particular discrete operator [34]. The resultingschemes are not only preferable as solvers in a multigrid strategy but are also su-perior single-grid schemes, in comparison to other explicit marching schemes, aspreconditioning already accelerates the convergence to a steady solution and the42

43high-frequency damping provides robustness.Two other advantages of local preconditioning are:� Local preconditioning inspires spatial discretizations that may retain their ac-curacy at low Mach number, in contrast to standard discretizations [33, 16, 58].� The local-preconditioning matrix can be designed so as to decouple the acousticequations from the convective equations in the system, making it possible togenerate genuinely multi-dimensional discretizations [38].This thesis focuses on the convergence-acceleration properties of local precondi-tioning. In particular, we will concentrate on the family of preconditioning matricesderived by Van Leer et al. [63, 32, 33]. Our focus is also on explicit time-steppingformulations. (For implicit formulations cf. [16]).3.1 Three Ways of Time SteppingThere are three distinct ways of marching in time with an explicit method.1. Global time-stepping. The same time-step value (�t) is used in all com-putational cells. This method yields time-accuracy of the numerical solution.However, convergence to the steady state is likely to be limited by sti�ness dueto spatial variations in cell size and the largest characteristic speed.2. Local time-stepping. Each cell is given its own time-step, scaling so asto yield a uniform Courant number for all cells. Time accuracy is lost, butconvergence to the steady state is accelerated and schemes can be designedsuch that the steady state solution is not a�ected.Local time-stepping is equivalent to preconditioning the residual by a scalar;the preconditioning removes the sti�ness due to spatial variations in cell size

44and the largest characteristic speed. The remaining is done to the spreadamong the local characteristic speeds.3. Characteristic time-stepping. Each characteristic variable in each cell isupdated with its own time-step. Even more time accuracy is lost, but conver-gence to the steady state, especially to subsonic and transonic solutions, maybe accelerated dramatically.This is equivalent to preconditioning the residual by a local matrix; it takesaway the sti�ness due to the variation among the local characteristic speeds,while the e�ect of cell aspect-ratio can also be neutralized.Characteristic time-stepping should be used together with local time-stepping formaximum e�ect.3.2 Preconditioning the One-Dimensional Euler EquationsThe construction of an e�ective matrix-preconditioner for the one-dimensionalEuler equations is relatively simple because the characteristic speeds and directionsare known without ambiguity.Consider the quasi one-dimensional Euler equations (e.g. ow in a channel ofvariable cross-section):@U@t = �A(U)@U@x + S(U; x) = Res(U) (3:1)As mentioned in the previous section, a local time-step is meant to achieve thesame Courant number (�) in every cell, i.e.(�t)i = � �xjqj+ c!i

45in cell i, where �xi is the length of cell i, and jqj+ c corresponds to the speed of thefastest moving wave, the forward acoustic wave. Using local time-stepping can beviewed as using global time-stepping for the Euler equations preconditioned in cell iby the scalar factorPi = �ti�t = �xjqj+ c!i24minj �xjqj+ c!j35�1 � 1where �t represents a globally-stable time-step. For a uniform grid, and using acontinuum extension of its de�nition, we may write the preconditioning factor asP = maxx(jqj+ c)jqj+ c = maxx �(A)�(A) ;where �(A) is the spectral radius of A. The preconditioned equations for localtime-stepping are therefore given by@U@t = P Res(U) = maxx �(A)�(A) Res(U):Using characteristic time-stepping means locally achieving the same Courant numberfor each characteristic equation, i.e.(�tk)i = � �xjakj!i; k = 1; 2; 3:For a uniform grid, and in continuum notation, characteristic time-stepping is equiv-alent to global time-stepping for the Euler equations preconditioned by the matrixP = �(A)jAj�1i.e., @U@t = P Res(U) = �(A)jAj�1Res(U); (3:2)where jAj is the matrix with the same eigenvectors as A but with absolute eigenval-ues.

46In practice, each of the eigenvalues of A may vanish locally, or be very small,making the inversion of jAj impossible, or undesirable. Equation (3.2) should bepreconditioned as @U@t = P Res(U) = �(A)jA�j�1Res(U); (3:3)where the eigenvalues of jA�j are bounded away from zero in the same way as is doneto avoid expansion shocks (Equation (2.45)).This preconditioning achieves a condition number of 1, independent of the localMach number (ignoring the e�ect of bounding the eigenvalues away from zero). Allcharacteristic speeds are made equal to the maximum speed (jqj + c) in absolutevalue.Preconditioning by jA�j�1 does not change the signs of the characteristic speeds,and thus does not interfere with the imposition of boundary conditions at eitherend of the channel. In consequence, the steady-state solution admitted by theseboundary conditions is also not altered.When characteristic time-stepping is used in conjunction with local time-steppingon a uniform grid, this is equivalent to using global time-stepping for the Eulerequations preconditioned by the matrixP = �maxx �(A)� jAj�1:The reader is referred to [33] for numerical convergence studies based on thisanalysis.3.3 Two-Dimensional PreconditioningUnlike the one-dimensional Euler equations, which only describe three waves inone dimension, the two-dimensional Euler equations admit an in�nite number of

47waves in a two-dimensional plane.It has been found advantageous for the development of preconditioners to intro-duce a coordinate system aligned with the instantaneous streamlines, and use theso-called symmetrizing variables, de�ned di�erentially by dUk = (dp�c ; dq; dr; ds)T ; sdenotes the entropy, i.e. ds � dp�c2d�. The system of Euler equations then becomes@ ~Uk@t = �Ak@ ~Uk@x �A? @ ~Uk@y ; (3:4)with Ak and A? symmetric and sparse. The derivation of such a preconditioningmatrix is fairly elaborate and will not be discussed here (cf. [33]). For the Eulerequations in this form, the preconditioner of Van Leer et al. [63] is given byP2D = 2666666666664 ��2M2 � ��2M 0 0� ��2M ��2 + 1 0 00 0 � 00 0 0 1 3777777777775 ; (3:5)where M is the local Mach number and � and � are de�ned by:� = 8>><>>: p1 �M2; M < 1;pM2 � 1; M � 1; (3.6)� = 8>><>>: p1 �M2; M < 1;p1 �M�2; M � 1: (3.7)This matrix achieves what can be shown to be the optimal condition number forthe characteristic speeds, namely, 1=p1�M2 for subsonic ow and 1 for supersonic ow. This is a major improvement over the condition number before preconditioning,which equals (M + 1)=min(M; jM � 1j). In practical applications, M and p1�M2need to be bounded away from zero. This is described in more detail in Section 6.2(see also [33]).

48The origin of the wave-speed scaling withp1 �M2 for subsonic ow is the elliptic(rather than circular) shape of an acoustic wave-front admitted by the preconditionedequations; its minor axis is aligned with the ow direction. For M = 1, the frontdegenerates into a pair of point disturbances emitted normal to the ow direction.For M > 1 the two acoustic disturbances propagate along Mach lines (cf. [33]).The local preconditioning matrix described in [63] was derived on the basis of thedi�erential form of the Euler equations. It is not a priori clear that this matrix is bestsuited for preconditioning discretized Euler equations. It was found, in particular,that high-frequency damping could be improved by modifying the matrix such that,for the discrete spatial operator, the high-frequency symbols corresponding to thedi�erent waves overlap more completely. This modi�ed Euler-preconditioning matrixwas �rst described in [32]; in this thesis the change occurs in the value of � .Another consequence of discretization is the introduction of grid parameters, suchas the aspect ratio and orientation of quadrilateral cells. These too must show up inthe value of � .Both e�ects are discussed further in Section 3.9.3.4 Three-Dimensional PreconditioningIt turns out that some further deterioration of the characteristic condition numbermust be expected when stepping up from two to three dimensions. This deteriorationoccurs as a result of a new kind of shear wave that can exist in three dimensions. Thiswave mode, which rotates the ow velocity, cannot be separated from the acousticwaves when algebraically manipulating the Euler equations. In consequence, scalingthe acoustic waves down to the ow speed for the supersonic case has the undesir-able e�ect of slowing down the 3-D shear wave as well. The optimal characteristic

49condition-number thus becomes 1=q1 �min(M2;M�2) for all M. To improve uponthis for supersonic ow, di�erential preconditioning is needed [51]. Only combina-tions of derivatives of the Euler equations can isolate the convection equation forstreamwise vorticity, so that it can be left untouched.The reader is referred to [33] for a detailed analysis of the three-dimensionalcase. We will limit ourselves to two-dimensional preconditioning for the rest of ourdiscussion.3.5 Implementing Characteristic Time-SteppingThe new set of equations obtained by preconditioning the system with the matrixde�ned above yield acoustic wave speeds that are almost independent of the directionof wave propagation (except for M " 1) and that equal the ow speed. Transientsolutions of these equations will di�er dramatically from those of the original Eulerequations. However, the steady form of these modi�ed equations is identical tothe steady Euler equations, so we would expect steady solutions of the two sets ofequations to be the same. This justi�es the use of the preconditioned equations innumerical applications. The more uniform wave speeds will lead to more e�cientmarching to steady solutions.Solving the preconditioned equations with regular time-steps is equivalent tosolving the original equations with characteristic time-steps. In the latter case, thepreconditioning matrix must be given the physical dimension of a time step. Char-acteristic time-stepping should be used together with local time-stepping for themaximum e�ect.Recall that the preconditioned two-dimensional Euler equations were derived ina stream-aligned coordinate system with a non-conserved set of ow variables. In

50order to be applied in a numerical scheme the residual must be expressed in terms ofthe conserved variables, as a ux divergence, and preferably in a Cartesian referencecoordinate system.The residual is �rst transformed to the stream-aligned coordinate system andvariables used in the preconditioning analysis. The solution is then advanced to a newtime level with local preconditioning using this transformed residual. This updateto the solution is then transformed back to the original variables and coordinatesystem and used to advance the solution. This procedure can be expressed in termsof matrix multiplications and is shown below.Consider the semi-discretized two-dimensional Euler equations, which can bewritten as @U@t = Res(U) (3:8)where the residual Res(U) is the numerical representation of the spatial operator ofthe original two-dimensional Euler equations.To transform the residual to stream-aligned coordinates and non-conservativevariables, Equation 3.8 is multiplied by Q�1M�1, where Q, Q�1 are transforma-tions between Cartesian coordinates and ow-aligned coordinates, and M, M�1 aretransformations between the di�erent variable systems:d~U =MdUk;dUk = Qd~U:Q and M are given in the next section. This gives@ ~Uk@t = Q�1M�1Res(U)The increment to the next time level is obtained by one characteristic time-step�~Uk = h ~�tiQ�1M�1Res(U);

51where the matrix time-step is related to the preconditioning matrix P2D (Equation3.5) by h ~�ti = �lP2Dwhere � and l are local values of Courant number and cell size. This increment istransformed back to Cartesian coordinates and conservative variables, yielding�U = MQ h ~�tiQ�1M�1Res(U) (3.9)= [�t]Res(U); (3.10)where [�t] =MQ h ~�tiQ�1M�1 (3:11)is the characteristic time-step for Cartesian coordinates and conservative variables.The de�nitions of � and l are not trivial and will be given in Section 4.4.3.6 Reformulation of Numerical FluxesWe are interested in discretizations that are based on upwind-biased spatial di�er-encing, combined with multi-stage timemarching. Van Leer et al. [63] have indicatedthat, in order to avoid a severe stability restriction on the time-step, the precondi-tioned scheme must have the form of a �rst-order upwind-di�erencing scheme for thepreconditioned Euler equations. The modi�cation amounts to a subtle change in thearti�cial-dissipation matrix for Roe's ux function [49]. In particular, the familiardissipation matrix jAj must be replaced by P�1jPAj. Preconditioning removes thefactor P�1, yielding an arti�cial dissipation matrix jPAj appropriate for a hyperbolicsystem with coe�cient PA. In contrast, a scalar dissipation term needs no modi�-cation. Furthermore, a di�erent preconditioner such as block-Jacobi may work wellwithout any modi�cation of the dissipation matrices [2].

52Consider again the two-dimensional Euler equations in a Cartesian coordinateframe: @U@t = �@F(U)@x � @G(U)@y = Res(U); (3:12)U = (�; �u; �v; �E)T ;where U, F and G represent the conserved state quantities and their uxes in the x-and y-directions. Recall that the residual computed by integration over a (quadri-lateral) �nite-volume cell can be written asResi;j = � 1Ai;j 4Xk=1 f�k�Skgi;j; (3:13)where Ai;j = area of cell (i; j);�Sk = length of the kth cell face;�k = ux normal to kth cell face:For a grid-aligned upwind di�erence scheme, the normal ux can be computed as� = F cos � +G sin �; (3:14)where � is the angle that the cell-face makes with the x-axis. The Roe upwind-biased ux in the �-direction at the interface of cells L (left state) and R (right state) isgiven by �Roe = 12(�L +�R)� 12 jj(UR �UL); (3:15)where is the Roe-average of the Jacobian of � for the pair (UL;UR).3.6.1 Roe's arti�cial-viscosity matrixThe original Roe ux has an arti�cal viscosity matrix that is given byj1j = jA cos � + B sin �j (3:16)

53where A and B are the Roe-averaged ux-Jacobians of F and G.Alternatively, we may solve two Riemann problems based on (UL;UR), one inthe direction of the ow and the other in the direction normal to it, and then mergethe two numerical uxes into one ux normal to the cell face; in this way we getj2j =MQ(jAkjj cos(�k � �)j+ jA?jj cos(�? � �)j)Q�1M�1; (3:17)where �k, �? are the ow angle and normal-to- ow angle. This matrix is much easierto use in combination with preconditioning and is identical to Equation (3.16) if the ow is in the x- or y-direction.jAkj and jA?j are given in symmetrizing variables asjAkj = c2666666666664 �2+M+12 ��2+M+12 0 0��2+M+12 �2+M+12 0 00 0 M 00 0 0 M 3777777777775 ; (3:18)jA?j = c2666666666664 1 0 0 00 0 0 00 0 1 00 0 0 0 3777777777775 : (3:19)3.6.2 Modi�ed Roe arti�cial-viscosity matrixFor the preconditioned scheme, the arti�cial-viscosity matrix j2j is replaced byjmodj =MQ(P�12DjP2DAkjj cos(�k � �)j+ P�12DjP2DA?jj cos(�? � �)j)Q�1M�1:(3:20)P�12DjP2DAkj and P�12DjP2DA?j can be written in terms of symmetrizing variables

54(dUk = (dp�c ; dq; dr; ds)T ):P�12DjP2DAkj = c2666666666664 �2+1M 1 0 01 M 0 00 0 M 00 0 0 M 3777777777775 ; (3:21)P�12DjP2DA?j = c2666666666664 �M 0 0 00 0 0 00 0 M� 00 0 0 0 3777777777775 : (3:22)The rotation matrices Q and Q�1 are given byQ = 2666666666664 1 0 0 00 cos �k � sin�k 00 sin�k cos�k 00 0 0 1 3777777777775 ; (3:23)Q�1 = 2666666666664 1 0 0 00 cos�k sin�k 00 � sin�k cos�k 00 0 0 1 3777777777775 : (3:24)These matrices transform the system between Cartesian and streamline coordinatesystems.The matrices M and M�1, which transform the system between conservative

55variables and symmetrizing variables, are given byM = 2666666666664 �c 0 0 � 1c2�uc � 0 � uc2�vc 0 � � vc2�c �12M2 + 1 �1� �u �v �12M2 3777777777775 ; (3:25)M�1 = 2666666666664 �12 cM2� �( � 1) u�c �( � 1) v�c �1�c�u� 1� 0 0�v� 0 1� 0c2 � �12 M2 � 1� �( � 1)u �( � 1)v � 1 3777777777775 : (3:26)Finally, the modi�ed upwind ux-function is given by�modRoe = 12(�L +�R)� 12 jmodj(UR �UL)Note that1. The expressions for jAkj and P�12DjP2DAkj are di�erent for M < 1, i.e., whenthe eigenvalues of jAkj have mixed signs.2. The expressions for jA?j and P�12DjP2DA?j are di�erent for all M since theeigenvalues of jA?j always have mixed signs.3.7 Fourier Footprints of the Spatial Euler OperatorFourier footprints, a visual representation of the spatial operator, are an impor-tant tool in the design and analysis of schemes for partial di�erential equations. Theyhave been used with success, for example, in the design of multi-stage time-steppingschemes [23, 66, 64, 34]. They are also useful in the context of preconditioning, asthey allow the study of the e�ect of preconditioning on a particular discrete spatialoperator.

56The Fourier footprint of a spatial-di�erencing operator is obtained by taking theFourier transform of the spatial operator and plotting its eigenvalues in the complexplane. The Fourier footprint of the operator is the locus of these eigenvalues in thecomplex plane.We shall now obtain the Fourier transforms for the �rst-order and higher-orderupwind discretizations of the preconditioned Euler equations using the modi�ed Roespatial operator.Consider the linearized Euler equations discretized on a uniform rectangularmesh, with cell dimensions �x and �y. We will assume that the system is rep-resented in symmetrizing variables; this does not a�ect the Fourier footprint. Wewill also use P to denote P2D from now on.3.7.1 Fourier footprint of the �rst-order upwind schemeWith characteristic time-stepping we obtain the following discretization for the�rst-order upwind scheme on a Cartesian grid using the modi�ed Roe ux function[�t]Res(U) = � �ljqjQPQ�1 ( A2�x�xU� 12�xQ �P�1jPAk cos �kj+ P�1jPA? sin�kj�Q�1�x�xU+ B2�y �yU� 12�yQ �P�1jPAk sin�kj+P�1jPA? cos �kj�Q�1�y�yU) ;(3.27)where �(:) is a central-di�erencing operator, l is a characteristic length-scale (cellwidth), � is the Courant number and jqj is the ow-speed.Inserting Fourier data into the scheme, of the formU = U0ei(�xx�x +�yy�y );

57where �x and �y are spatial frequencies, we obtainFT ([�t]Res) = � �ljqjQPQ�1 (i A�x sin�x + B�y sin�y!+ 1�xQ �P�1jPAk cos �kj+P�1jPA? sin�kj�Q�1(1� cos �x)+ 1�yQ �P�1jPAk sin�kj+P�1jPA? cos �kj�Q�1(1� cos �y)) :(3.28)In this special case the eigenvalues of the Fourier transform can be obtainedanalytically; in general we shall make use of numerical software to compute theeigenvalues, viz. EISPACK [53].3.7.2 Fourier footprints of the � family of upwind schemesThe Fourier footprints of the � schemes with the modi�ed Roe ux function andcharacteristic time-stepping may similarly be obtained as:FT ([�t]Res�) = � �ljqjQPQ�1 "i( A2�x �1� �2 sin 2�x � (3� �) sin�x�+ B2�y �1� �2 sin 2�y � (3 � �) sin �y�)+ Q2�x �P�1jPAk cos �kj+P�1jPA? sin�kj�Q�1�(2(1� �) cos �x � 1� �2 cos 2�x � 3(1� �)2 )+ Q2�y �P�1jPAk sin�kj+P�1jPA? cos�kj�Q�1�(2(1� �) cos �y � 1 � �2 cos 2�y � 3(1 � �)2 )# : (3.29)3.8 E�ect of Preconditioning on the Spatial OperatorIn the two-dimensional Euler equations there are di�erent kinds of physical sig-nals propagating in all possible directions at di�erent speeds; these are more or less

58�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 0:1, � = 0o, � = 1Fourier Footprint of First-Order Roe SchemeFigure 3.1: Fourier footprint of the �rst-order upwind approximation of the spatialEuler operator, for M = 0:1, and ow angle � = 0�. The time-stepchosen corresponds to a Courant-number value of 1.accurately represented by the discrete operator and produce di�erent concentrationsof eigenvalues in its Fourier footprint.Figures 3.1 to 3.4 show the Fourier footprint of the �rst-order upwind schemefor the Euler equations, based on Roe's upwind-biased ux formula, for a range ofMach numbers. All �gures are for the case when the ow speed is aligned with thegrid (� = �k = 0) and the cell aspect-ratio is 1. The di�erent sizes of the di�erentconcentrations in the footprint make it impossible to place the zeros of a multi-stageampli�cation factor at �xed locations in the complex plane and still achieve goodhigh-frequency damping for all Mach numbers (even disregarding ow angles). Thefrequencies included in these and any further footprints are �x 2 [0; �], �y 2 [0; �],rather than �x 2 [��; �], �y 2 [��; �], to prevent cluttering of the �gure. With thischoice, eigenvalues in the upper/lower half plane mostly correspond to waves movingupstream/downstream.

59�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 0:5, � = 0o, � = 1Fourier Footprint of First-Order Roe SchemeFigure 3.2: Fourier footprint of the �rst-order upwind approximation of the spatialEuler operator, for M = 0:5, and ow angle � = 0�.

�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10<(z)=(z) M = 0:9, � = 0o, � = 1Fourier Footprint of First-Order Roe Scheme

Figure 3.3: Fourier footprint of the �rst-order upwind approximation of the spatialEuler operator, for M = 0:9, and ow angle � = 0�.

60�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 2:0, � = 0o, � = 1Fourier Footprint of First-Order Roe SchemeFigure 3.4: Fourier footprint of the �rst-order upwind approximation of the spatialEuler operator, for M = 2, and ow speed aligned with the grid.

�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10<(z)=(z) M = 0:1, � = 0o, � = 1First-Order modi�ed Roe Scheme

Figure 3.5: Fourier footprint of the preconditioned �rst-order upwind approximationof the spatial Euler operator, for M = 0:1, and ow angle � = 0�.

61�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 0:5, � = 0o, � = 1First-Order modi�ed Roe SchemeFigure 3.6: Fourier footprint of the preconditioned �rst-order upwind approximationof the spatial Euler operator, for M = 0:5, and ow angle � = 0�.

�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10<(z)=(z) M = 0:9, � = 0o, � = 1First-Order modi�ed Roe Scheme

Figure 3.7: Fourier footprint of the preconditioned �rst-order upwind approximationof the spatial Euler operator, for M = 0:9, and ow angle � = 0�.

62�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 2:0, � = 0o, � = 1First-Order modi�ed Roe SchemeFigure 3.8: Fourier footprint of the preconditioned �rst-order upwind approximationof the spatial Euler operator, for M = 2, and ow angle � = 0�.The next sequence of �gures, Figures 3.5 to 3.8, show the Fourier footprint ofthe preconditioned �rst-order upwind scheme for the Euler equations, again basedon Roe's ux formula, for a range of Mach numbers. The preconditioning matrix isthe one presented in [63] and in Equations (3.5 - 3.7). A comparison with the previ-ous sequence shows that removing the variation among the characteristic convectionspeeds has resulted in a thorough clean-up of the footprint. The e�ect of the precondi-tioning matrix is especially impressive for small M . For M " 1 a growing separationof two regions of concentration of eigenvalues is observed; this corresponds to thegrowing disparity between the acoustic speeds in the ow direction (= qp1 �M2,q � ow speed) and normal direction (= q). For M > 1 the footprint starts lookingvery much like one for scalar convection (see Figure 4.3); this is because all signalsin supersonic ow move downstream.

63�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 0:1, � = 0�, � = 2First-Order modi�ed Roe SchemeFigure 3.9: Fourier footprint of the preconditioned �rst-order upwind approximationof the spatial Euler operator, for M = 0:1, and ow angle � = 0�.j�xj; j�yj 2 [0; �] : Courant number = 2. Modi�ed preconditioner (Section3.9) used here.3.9 Relocating High-Frequency Content in the FootprintThe local preconditioning matrix described in [63] was derived based on thepartial-di�erential equations (PDEs) that make up the Euler equations. PDE-basedanalysis can be expected to accurately predict the e�ect of the preconditioner onlow-frequency error components. However, high-frequency error components are verypoorly described by the PDEs. This may force changes in numerical ux functionsfor reasons of stability. The modi�ed Roe scheme [63] is one result. Similarly, asmentioned earlier, on analysis of the Fourier footprint of the discrete scheme [32],it was realized that high-frequency damping could be improved by modifying thepreconditioning matrix such that the high-frequency symbols due to all the di�erentwaves overlap more completely.If we allow � to vary as a free parameter, we can rescale the Fourier symbols cor-responding to the acoustic error-modes such that the high-frequency Fourier symbols

64(with j�xjand/orj�yj 2 (�2 ; �)) overlap with the high-frequency symbols correspond-ing to the shear and entropy error-modes. This allows the design of multi-stageschemes with improved damping of high-frequency error-modes.In a manner similar to the analysis of D. Lee, [31], we look at the Fourier sym-bols corresponding to the high-high spatial frequency point �x; �y = �. This pointcorresponds to the Fourier symbol with largest real part. The Fourier symbols arealso purely real for this frequency point, making analysis easier. For the �rst-orderupwind scheme with modi�ed Roe ux-function (Equation 3.28) we obtainFT(�;�)([�t]Res) = � 2�ljqj�xQnjPAk cos�kj+ jPA? sin �kj+AR �jPAk sin�kj+ jPA? cos �kj�Q�1o ; (3.30)where AR is the cell aspect-ratio (�x=�y).We obtain two sets of repeated eigenvalues from this matrix. These correspondto the (�; �) Fourier symbols of the two acoustic waves, which are identical forthis spatial-frequency combination and the (�; �) Fourier symbols for the shear andentropy waves, which are identical for all frequency pairs, at least with the Van Leerpreconditioner. The acoustic eigenvalues are given by�ac = � 2�ljqj�x (M� 1 + AR� ! j cos �kj+M� AR+ 1�! j sin �kj) : (3:31)The shear-entropy eigenvalues are given by�sh;en = � 2�ljqj�x nM �j cos�kj+ARj sin�kj�o : (3:32)D. Lee and Van Leer [32] suggest scaling up the shear-entropy point to matchthe acoustic high-high point. We propose scaling down the acoustic Fourier symbolsto match those of the shear-entropy Fourier symbols. This allows us to obtain a

65de�nition of the Courant number that is consistent with de�nitions for scalar equa-tions and that is also based on the ow-speed jqj. It also has the advantage ofconcentrating all the changes in P into the value of � .Equating the acoustic eigenvalues to the shear-entropy eigenvalues gives us thefollowing relation for � :� = �= j sin �kj+ARj cos �kjj cos �kj+ARj sin�kj + �! : (3:33)We de�ne ARq = j sin�kj+ARj cos �kjj cos�kj+ARj sin�kj ; (3:34)where ARq may be interpreted as an aspect ratio based on the cell dimensions in the ow and normal directions for a rectangular cell [31] (see also Figure 5.1). Insertingthis de�nition into Equation (3.33), we obtain� = �ARq + � : (3:35)For isotropic cells typical of Euler calculations, this simpli�es further to� = �1 + � : (3:36)Note that � is de�ned as before (Equation (3.5)). Scaling the Fourier symbolscorresponding to the � family of upwind schemes yields the same result (Equation(3.35)).Figure 3.5 shows Fourier footprint of the modi�ed Roe scheme at M = 0:1, ow-angle � = 0�, AR = 1, with the original preconditioner of Van Leer et al.Figure 3.9 shows the same case with the modi�ed preconditioner. Note the locusof entropy/shear eigenvalues, an arc in the lower half-plane, which has now becomevisible. The modi�ed Roe ux-function remains unchanged by this scaling.We will use this version of the preconditioner to design optimally damping multi-stage schemes.

66�6:00�5:00�4:00�3:00�2:00�1:00 0:00 1:00�3:50�2:50�1:50�0:500:501:502:503:50

<(z)=(z) M = 0:1, � = 45�, � = 2First-Order modi�ed Roe SchemeFigure 3.10: Fourier footprint of the preconditioned �rst-order upwind approxima-tion of the spatial Euler operator, for M = 0:1, and ow angle � = 45�.j�xj; j�yj 2 [0; �] : Courant number = 2. Modi�ed preconditioner (Sec-tion 3.9) used here.3.10 E�ect of the Flow Angle and Aspect Ratio on theFootprintFigure 3.10 shows the footprint under the same conditions as Figure 3.9, exceptthat the ow is no longer aligned with the grid (� = 45�). The main e�ect of thisrotation is that the entropy/shear eigenvalues no longer lie on a curve but �ll in anarea. The consequences of this behavior of the footprint for the design of optimallysmoothing multi-stage schemes will be discussed in Section 4.1.Figure 3.10 shows the change with respect to Figure 3.9 if only the aspect ratiois changed, viz. from 1 to 4. The emergence of a second length-scale with growingAR, i.e. �y = �x=AR, causes a separation of eigenvalue concentration similar to theone observed for M ! 1.When optimizing the smoothing properties of multi-stage schemes, dealing withthe disparate scales in the Fourier footprint caused by largeAR and/or smallp1�M2

67�4:20�3:60�3:00�2:40�1:80�1:20�0:60 0:00�2:10�1:50�0:90�0:300:300:901:502:10

<(z)=(z) M = 0:1, � = 0�, AR = 4, � = 2First-Order modi�ed Roe SchemeFigure 3.11: Fourier footprint of the preconditioned �rst-order upwind approxima-tion of the spatial Euler operator, for M = 0:1, and ow angle � = 0�.AR = 4. j�xj; j�yj 2 [0; �] : Courant number = 2. Modi�ed precondi-tioner (Section 3.9) used here.is not trivial. As explained in Sections 4.5-6, this situation calls for a more sophisti-cated multigrid technique, viz. semi-coarsening.

CHAPTER IVDESIGN OF OPTIMAL MULTI-STAGESCHEMESExplicit marching schemes are commonly used as multigrid relaxation schemeswhen solving the Euler and Navier-Stokes equations. In order to be suited for usein multigrid marching these schemes must feature e�ective high-frequency damp-ing. Multi-stage schemes o�er the exibility to damp certain (hopefully most) high-frequency waves admitted by a particular discrete operator.Until recently, however, the design of optimally smoothing multi-stage schemeswas based entirely on the scalar one-dimensional [23, 66] or two-dimensional [10]convection equation, even though the schemes were intended for use with Euler andNavier-Stokes spatial operators. Euler and Navier-Stokes spatial operators produceseveral concentrations of eigenvalues in their Fourier footprints, making it impos-sible to place the zeros of a multi-stage ampli�cation factor at �xed locations inthe complex plane and still achieve good high-frequency damping for all Mach num-bers. Fortunately, the local-preconditioning matrix of Chapter III has the e�ect oforganizing and concentrating the pattern of such eigenvalues, allowing the design ofmulti-stage schemes that systematically damp most high-frequency waves admittedby the particular discrete operator. 68

69The resulting schemes are not only preferable as solvers in a multigrid strategy,but are also superior single-grid schemes, in comparison to other explicit marchingschemes, as the preconditioning itself already accelerates the convergence to a steadysolution, and the high-frequency damping due to these schemes provides robustness.We will now describe some of the scalar analysis referred to earlier on, followedby some extensions of this scalar analysis. We will later consider extensions of thisanalysis to systems.4.1 Previous Scalar AnalysisWe will begin by describing some of the earlier design approaches that make useof the one-dimensional convection equation. Jameson's [23] approach is based ontrial and error. The �rst attempt at systematic optimization of multi-stage methodswas by Smith and Caughey [54]. A similar, but more comprehensive, approach isthat of Tai et al. [66, 56]. Tai's procedure for optimizing the high-frequency dampingin a one-dimensional convection scheme is a geometry exercise in the complex plane:putting the zeros of the multi-stage ampli�cation factor on top of the locus of theFourier transform - the \Fourier footprint" - of the discrete spatial operator. Thisplacement of zeros can be achieved for one speci�c value of the time-step; obtainingthis value is part of the design process. An example of the result of this procedure isshown in Figures 4.1 and 4.2, showing the footprint on top of level lines of the multi-stage ampli�cation factor, and the ampli�cation factor as a function of frequencyrespectively. Catalano and Deconinck [10] relaxed the condition that the zeros mustlie exactly on the Fourier locus, thereby achieving a further reduction of the maximumampli�cation factor for the high frequencies in some cases. Results identical to thosein [66] were obtained for �rst-order upwind and � = �1 discretizations; a � = 1=3

70�9:0 �7:0 �5:0 �3:0 �1:0 1:0�5:0�3:0�1:01:03:05:0 <(z)=(z) Six-Stage SchemeFourier Footprint of � = 1=3 1 1.002 0.853 0.654 0.505 0.356 0.207 0.058 0.01

Figure 4.1: Fourier footprint (dashed line) of the third-order upwind-biased spa-tial discretization of the one-dimensional convection operator, and levellines (solid) of the ampli�cation factor of Tai's optimal six-stage scheme.(Design-graph of Tai's optimal six-stage scheme).discretization was shown to produce di�erent coe�cients with a lower value of themaximum ampli�cation factor.The Fourier footprint for a two-dimensional discrete convection operator is nolonger a single curve, but covers an area; the location and shape of this area varygreatly with the convection direction. Figures 4.3 and 4.4 show the locus for the�rst-order upwind-di�erencing operator for convection directions of 10� and 45�. Thespatial frequencies included in the footprint are �x 2 [0; �], �y 2 [0; �], which are asin Section 3.8.For the two-dimensional discrete operator, damping of high-frequencies is easilyachieved by a �xed multi-stage scheme (coe�cients independent of ow direction) formodes propagating in the physical convection direction, but is fundamentally di�cultfor modes varying in the normal direction, especially if the convection is almost in thegrid direction. This is the single-grid alignment problem. The alignment problem

710:00 �=4 �=2 3�=4 �0:000:250:500:751:00

�jP j Six-Stage SchemeMagnitude of Ampli�cation Factor for � = 1=3Figure 4.2: Modulus of the ampli�cation factor as a function of spatial frequency,for the case of Figure 4.1.is evident in Figure 4.3 from the low-high frequency combinations found near theorigin. To damp these frequency combinations, zeros must be put close to the origin,making it necessary to have a large time-step in order to bene�t from these zeros.This works against numerical stability.For two-dimensional convection, Tai, as well as Catalano and Deconinck, use aone-dimensional optimization: they only consider high-frequency plane waves movingin the ow direction. Tai accepts the optimal sequence of time-step ratios for one-dimensional convection and merely adjusts the �nal Courant number; the latterdepends on the ow angle. Catalano and Deconinck repeat the optimization foreach ow angle; however this makes all parameters dependent on the ow angle,which is less desirable. Moreover, the alignment problem causes the parameters tovary strongly when the ow angle becomes small: the optimization problem becomespoorly formulated.

72�3:25�2:75�2:25�1:75�1:25�0:75�0:25 0:25�1:75�1:25�0:75�0:250:250:751:251:75

<(z)=(z) � = 10oFourier Footprint of First-Order Upwind SchemeFigure 4.3: Fourier footprint of the �rst-order upwind approximation of the two-dimensional convection operator; convection angle � = 10�.4.2 Extensions of Scalar AnalysisAll the analysis described thus far has made use of a one-dimensional or a reduced-dimension two-dimensional Fourier footprint. It is fairly simple to formulate an opti-mization problem based on the entire Fourier footprint, as will be described later onin this chapter. The optimization problem would be ill-de�ned if we did not excludecertain low-high frequency combinations for small ow angles (a manifestation ofthe single-grid alignment problem). Our formulation therefore excludes these com-binations by means of a \wedge" �lter. Details of the formulation are described inSection 4.3 of this chapter.We have computed multi-stage coe�cients based on an optimization of the en-tire two-dimensional operator. It turns out that the coe�cients obtained from thisoptimization are very close to those obtained from an optimization based on thefootprint of the full preconditioned Euler equations, discretized using the modi�ed

73�3:25�2:75�2:25�1:75�1:25�0:75�0:25 0:25�1:75�1:25�0:75�0:250:250:751:251:75

<(z)=(z) � = 45oFourier Footprint of First-Order Upwind SchemeFigure 4.4: Fourier footprint of the �rst-order upwind approximation of the two-dimensional convection equation; convection angle � = 45�.Roe ux function. This is indicated in Figure 4.5. The footprint in this �gure isfor the �rst-order upwind convection operator; the ampli�cation level lines are for a4-stage scheme optimized for the �rst-order upwind Euler operator. The level linesclosely follow the shape of the footprint; this is because the Euler footprint is domi-natied by the convective entropy/shear eigenvalues. The contours in the �gure comefrom the tables of optimal coe�cients that will be presented later in the chapter.These tables may therefore be used with schemes for the two-dimensional convec-tion equation even though they were derived based on footprints of a preconditionedEuler operator. More importantly, the coe�cients could have been derived on thebasis of the convection operator after all. However, this is only true when the entirehigh-frequency domain is being considered in the optimization. High-high frequencyacoustic eigenvalues also a�ect the coe�cients obtained in the optimization whenonly the high-high frequency domain is considered (cf. Section 4.6).Our experiments indicate that there is almost no variation in the values of multi-

74�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) 4-stage, First-order upwindFourier Footprint, 2-D scalar advection 1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.02 10Figure 4.5: High-frequency Fourier footprint of the �rst-order upwind 2-D scalar ad-vection operator plotted on top of the level lines of the ampli�cationfactor of the 4-stage scheme optimized using the modi�ed Roe operatorfor the Euler equations. Convection angle 45�.stage coe�cients with ow angle, for a given two-dimensional convection spatialoperator. The Courant number � does vary signi�cantly though. It is possible tomake an appropriate choice of the length scale used in de�ning the Courant numberand thus remove this ow angle variation (cf. Section 4.4).In contrast, large values of the aspect ratio and/or small values of p1 �M2 makethe optimization ill posed when the entire high-frequency domain is considered. Theproblem disappears when only the high-high frequency domain is considered; the re-sulting schemes are useful only if combined with semi-coarsened multigrid relaxation(cf. Section 4.6).We will now describe the formulation of the optimization problem in detail.

754.3 Formulation of the Optimization ProblemThe procedure for optimizing high-frequency damping aims at minimizing themaximum of the modulus of the scheme's ampli�cation factor over the set of high-frequency eigenvalues. The input parameters are the time-step values �t(k), k =1; ::;m; of an m-stage algorithm. When updating the solution ofUt = Res(U) (4:1)from time level tn to tn+1 = tn +�t, the multi-stage method takes the formU(0) = Un; (4.2)U(k) = U(0) +�t(k)Res �U(k�1)� ; k = 1; ::;m; (4.3)Un+1 = U(m) (4.4)with �t = �t(m): According to linear theory, one step with the full schememultiplieseach eigenvector of the operator Res(U), with associated eigenvalue �, by a factorof the form P (z) = 1 + z + mXk=2 ckzk; (4:5)where z = ��t (4:6)is generally complex. The m � 1 coe�cients ck relate to the time-step ratios �k =�t(k)=�t; the actual time step �t is the mth parameter.To obtain the functional that needs to be minimized, one starts out by com-puting a discrete set of eigenvalues for frequency pairs (�x; �y) in the high-frequencyrange. This set of eigenvalues corresponds to a �xed combination of other parametersthat the spatial operator depends upon, speci�cally, convection angle � for the two-dimensional convection equations; Mach number M and ow angle � for the Euler

76

�� �x���

���y

Figure 4.6: Wedge-�lter that is used to make the optimization problem meaning-ful for ow-angles near 0 (or �=2). The shaded region represents theportion of the high-frequency domain that is considered; its boundaryis controlled by a \wedge" drawn in the central, low-frequency domain.In this case, � < �=4 and the wedge angle is given by = � + �=4.For �=4 � � � �=2, the shaded region would rotate by �=2, such thathigh-frequency combinations around the �x axis are excluded.

77equations; and M , �, aspect ratio AR and Reynolds numberRe for the Navier-Stokesequations. We shall restrict ourselves presently to Euler discretizations; note thatfor Euler grids we shall assume that AR = 1 is a good approximation.The high-frequency domain may be de�ned asD = �(�x; �y) : j�xj 2 ��2 ; �� and=or j�yj 2 ��2 ; ��� : (4:7)if the multigrid scheme does not involve semi-coarsening [40].Assuming a set of starting values for the m-stage scheme, for instance Tai'svalues, the value of jP (z)j is computed for all eigenvalues previously obtained, andits maximum is found. This is our functional �(�t(1); ::;�t(m);M;�); it must beminimized by varying the m parameters. Dependence on M and � will be consideredlater.The optimal (in the L1 sense) m-stage scheme may hence be obtained as thesolution to the following minmax problem:�opt = min(~�;�) max(�x;�y)2D kP (z(�x; �y; �); ~�)k! : (4:8)The alignment problem (cf. Section 2.3,) makes the optimization meaninglessfor ow angles near 0 (or �=2), since the ampli�cation factor for low-high (or high-low) frequency combinations tends to 1 in this case. Our solution is to �lter outthese frequency combinations, e.g. for � = 0 we optimize only over the eigenvalueswith j�xj � �=2. A wedge drawn in the central, low-frequency domain controlsthe boundary of the optimization domain; see Figure 4.6. When � increases, thewedge opens up, until for � = �=4 the entire domain (Equation 4.7) is used; When� increases further, the optimization domain shrinks again, until for � = �=2 onlythose frequency pairs are included satisfying j�yj � �=2. The alignment problem

78itself cannot be solved by any improvement of the multi-stage schemes; it has to bedealt with separately. In this thesis it is solved by semi-coarsening.If the multi-stage scheme is to be used in conjunction with a semi-coarseningmultigrid algorithm, then only high-high frequency pairs need be considered. Thiscorresponds to D = �(�x; �y) : j�xj 2 ��2 ; �� and j�yj 2 ��2 ; ��� : (4:9)In this case the optimization problem is well-formulated and there is no need to makeuse of the wedge-�lter described above.4.4 Dependence on the Flow AngleAs explained above, the optimization procedure generates a time-step value forwhich the optimal high-frequency damping is realized. For a given spatial operatorand number of stages, this �topt depends on the Mach number and the ow angle.The variation with the ow angle is similar to that of the maximum permitted timestep �tmax for explicit convection schemes: on a square grid the stability limit dropsby a of factor p2 when the ow angle varies from 0� to 45�. This factor can beremoved by rede�ning the Courant number.For the preconditioned Euler equations, with the characteristic speeds equal toor close to q, we de�ne the Courant number as� = q �tl(�x;�y; �); (4:10)where l is a typical cell-width that may depend on the ow direction. Figure 4.7shows the typical variation of �opt with the ow angle for a square grid, using a �xedl = �x = �y. For small � the optimal value of the Courant number is dictated bythe size of the acoustic footprint; between 10� and 15� the entropy/shear footprint

790: 20: 40: 60:1:001:201:401:60

��=�45 Variation of Courant number with ow angle.1st order modi�ed Roe spatial operatorM = 0:1M = 0:3M = 0:5M = 0:7M = 0:9Figure 4.7: Variation of the optimal Courant number with the ow angle, forthe �rst-order upwind 4-stage scheme. Optimization over entire high-frequency range minus region removed by wedge �lter (see Section 4.3).takes over. (This feature is a function of the wedge �lter in the frequency domain,described in Section 4.3; it does not arise with semi-coarsening). The curves fordi�erent Mach numbers are very close, except when M approaches 1 (cf. Section4.5). For general rectangular cells we �nd that de�ningl = �x=(j cos�j+ARj sin �j) (4:11)takes away most of the variation of �opt with the ow angle, so that a single value canbe recommended. The geometrical interpretation of this formula is given in Figure4.8, due to Tai [56]; it is the distance of a cell corner to the opposite cell-diagonal,measured in the stream-wise direction.Another way of writing Equation (4.11) isl = �x�y�yj cos�j+�xj sin�j; (4:12)i.e., l equals the area of the cell divided by its dimension in the direction normal to

80�yA �x tan�BCD l E�xFigure 4.8: A geometrical interpretation of Equation (4.11). l � AE is the distancefrom a cell corner to the opposite cell-diagonal.the ow.4.4.1 Obtaining a de�nition for the Courant number length-scaleEquation (4.11) may be obtained analytically as follows. The variation in theoptimal multistage coe�cients with ow angle is an e�ect of the variation in the shapeof the Fourier footprint of the underlying discrete spatial operator. The minmaxformulation of the optimization problem turns out to be dependent primarily on theoutermost Fourier symbols in the footprint. Studying the change with ow anglein the symbols corresponding to the (�; �) frequency point would therefore indicatesome sort of scaling for the variation in coe�cients.To achieve no variation with ow angle in the eigenvalues corresponding to the(�; �) frequency point (Equations (3.31) and (3.32)), we absorb the variation with ow angle of the eigenvalues into the length scale, leading to the conditionl�x (j cos �j+ARj sin�j) = 1: (4:13)This leads to the de�nitionl = �x=(j cos �j+ARj sin �j):

81This scaling has been used in our numerical studies (Chapter VI).4.5 Optimization over Entire High-Frequency DomainWe present the sequence of �gures 4.9 - 4.12 to show what the technique describedcan accomplish. These all include the Fourier footprint of the spatial operator plottedon top of the level lines of the ampli�cation factor of the multi-stage scheme that hasbeen optimized for use with this operator. We will refer to these �gures as \design-graphs" from now on. The acoustic eigenvalues in these footprints are representedas plus signs (+) and shear/entropy eigenvalues by � signs. The eigenvalues in thefootprint are computed numerically, making it impossible to know a priori which ofthe four values corresponds to each of the di�erent waves. The entropy eigenvaluesare always known analytically, since the entropy convection-equation and its dis-cretization are fully decoupled from the other equations. Next we use the propertythat the shear and entropy eigenvalues are equal with this preconditioner, in order toseparate the convective eigenvalues from the acoustic eigenvalues. (Similar proper-ties may be used with other preconditioners in order to separate out the eigenvaluescorresponding to the di�erent waves).The size of the footprint scales with �t; optimal high-frequency damping isachieved only for one particular value of �t. How to obtain these results, i.e. howto actually solve the optimization problem is described in Appendix A; here we onlymention that we use a combination of the method of simulated annealing and thedownhill-simplex method of Nelder and Mead [46, 8].Figure 4.9 shows, for M = 1, � = 45�, how well the level lines can follow the out-line of the footprint of the �rst-order upwind Euler operator when 3 stages are used.The kidney-shaped outline is due to the high-frequency entropy/shear eigenvalues;

82the claw-shaped feature is the locus of the high-frequency acoustic eigenvalues. Thevalue of the functional is 0.3523, i.e., all high-frequency combinations are reducedat least by this factor. When adding one more stage (Figure 4.10), the attenuationis even better, namely, a factor of 0.2362; the �ve-stage scheme (Figure 4.11) yields� = 0:1652.Figure 4.12 shows how the footprint changes when the ow angle is reduced tozero: it becomes more transparent, but the outline stays the same. The level linesare those of the 4-stage scheme of Figure 4.10; the only change in the scheme is theadjustment of the time step (see section 4.4). It is seen that the functional hardlychanges (� = 0:2386) from the earlier cases considered; there is no need for furtheroptimization.Level lines of the ampli�cation factor in the (�x; �y) plane are shown in Figures4.13 - 4.16, all forM = 0:1. Data in the central square are suppressed: this is the low-frequency region. It must be remembered that each frequency combination createsfour eigenvalues: two are acoustic in nature, the other two correspond to entropyand shear. Each of these generates its own ampli�cation factor. In Figure 4.13, with� = 45�, for each combination of frequencies the maximum of the four ampli�cationfactors is plotted. In Figure 4.14 only the acoustic eigenvalues are included, whilein 4.15 only the entropy- and shear-related eigenvalues are included. Finally, inFigure 4.16, � = 0�, illustrating the alignment problem. Only the entropy/shear-related ampli�cation factor is plotted; the level lines show independence from �y.For �x = 0 the ampli�cation factor equals 1 regardless of the �y frequency. Thesefrequency combinations are excluded in the optimization procedure, as explainedearlier. The wedge-shaped optimization region is indicated by the dashed lines inthe �gure.

83�6:00�5:00�4:00�3:00�2:00�1:00 0:00 1:00�3:50�2:50�1:50�0:500:501:502:503:50

<(z)=(z) Optimal 3-stage, �(~�; �) = �opt = 0:35231st order modi�ed Roe Scheme,M = 0:1, � = 45�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.02 10Figure 4.9: Design-graph for the optimal 3-stage scheme based on the preconditioned�rst-order upwind Euler operator. Flow angle 45�.

�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:23621st order modi�ed Roe Scheme,M = 0:1, � = 45�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.02 10

Figure 4.10: As Figure 4.9, but for optimal 4-stage scheme.

84�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 5-stage, �(~�; �) = �opt = 0:16361st order modi�ed Roe Scheme,M = 0:1, � = 45�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.01 10Figure 4.11: As Figure 4.9, but for optimal 5-stage scheme.

�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25<(z)=(z) Optimized 4-stage, �(~�; �) = 0:23861st order modi�ed Roe Scheme,M = 0:1, � = 0�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.01 10

Figure 4.12: As Figure 4.10, but for ow angle 0�.

85�1:00 �0:50 0:00 0:50 1:00�1:00�0:500:000:501:00

�x=��y=� 1st order 4-stage, z from footprint of all wavesmax jP (z(�x; �y; �); ~�))j, M = 0:1, � = 45� 1 0.0222 0.0453 0.0674 0.0895 0.1116 0.1347 0.1568 0.1799 0.20110 0.2233 103 2 14 38 1010 2Figure 4.13: Level lines of the maximum ampli�cation factor in the (�x; �y) planeover the high-frequency domain. 1st order 4-stage scheme, � = 45�.

�1:00 �0:50 0:00 0:50 1:00�1:00�0:500:000:501:00�x=��y=� 1st order 4-stage, z from footprint of acoustic wavesContour plot of jP (z(�x; �y ; �); ~�))j, M = 0:1, � = 45�1 0.0222 0.0433 0.0654 0.0875 0.1096 0.1307 0.1528 0.1749 0.19610 0.2172 3410 115 10

Figure 4.14: As Figure 4.13, but for acoustic waves only.

86�1:00 �0:50 0:00 0:50 1:00�1:00�0:500:000:501:00

�x=��y=� 1st order 4-stage, z from footprint of entropy waveContour plot of jP (z(�x; �y; �); ~�))j, M = 0:1, � = 45�1 0.0222 0.0433 0.0654 0.0875 0.1096 0.1307 0.1528 0.1749 0.19610 0.21732 3213 101410 8 102 3Figure 4.15: As Figure 4.13, but for shear/entropy waves only.

�1:00 �0:50 0:00 0:50 1:00�1:00�0:500:000:501:00�x=��y=� 1st order 4-stage, z from footprint of entropy waveContour plot of jP (z(�x; �y ; �); ~�))j, M = 0:1, � = 0�1 0.0912 0.1823 0.2734 0.3645 0.4556 0.5467 0.6368 0.7279 0.81810 0.9092231022

Figure 4.16: As Figure 4.15, but for ow angle 0�. The dashed line outlines theportion of the domain considered (the portion containing the line �y =0).

87�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimized 4-stage, �(~�; �) = 0:26041st order modi�ed Roe Scheme,M = 0:5, � = 0�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.01 10Figure 4.17: Design-graph of the optimal 4-stage scheme based on the preconditioned�rst-order upwind Euler operator. Flow angle 0�, M = 0:5.4.5.1 Dependence on Mach NumberFigures 4.17 and 4.18 show the result of optimizing the 4-stage scheme for higherMach numbers than before, viz. M = 0:5 and M = 0:9; the ow angle is 0�. Theacoustic footprint bears evidence of a growing disparity among the characteristicspeeds: acoustic waves traveling in the ow direction only move at a speed qp1 �M2,while normal to the ow direction the propagation speed still equals q. The lowerspeed moves a group of high-frequency eigenvalues toward the origin, causing higherfunctional values for a given number of stages. In comparison to Figure 4.18 it isseen that � is increased only slightly (to 0.2604) for M = 0:5, but signi�cantly (to0.4100) for M = 0:9.It was our goal, given the spatial di�erencing operator and the number of stages,to produce a single set of multi-stage parameters that yield e�ective high-frequencydamping for any ow angle or Mach number. As explained in the previous section, the

88�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimized 4-stage, �(~�; �) = 0:41001st order Modi�ed Roe Scheme,M = 0:9, � = 0�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.02 10Figure 4.18: Design-graph of the optimal 4-stage scheme based on the preconditioned�rst-order upwind Euler operator. Flow angle 0�, M = 0:9.in uence of the ow angle is minor, once the alignment problem has been removed.By the same token, the in uence of the Mach number is minor once we recognizethere is no remedy for the deterioration of high-frequency damping as M approaches1, other than using more and more stages. Instead, for a �xed number of stages, wemay consider each set of optimal multi-stage parameters for M < 1 with a weightp1�M2. Thus, the sonic problem is removed, and a useful set of parameters canbe chosen. In practice this boils down to choosing a set of values for low M .Finally, there is no remedy for the deterioration of high-frequency damping forAR ! 1; this we have avoided by insisting of aspect-ratio values near 1 for Eulergrids.In Section 4.6 it will be shown that in using multigrid relaxation based on semi-coarsening, the above three problems (� ! 0� or 90�, M ! 1, AR ! 1) arecompletely avoided.

894.5.2 Optimal multi-stage coe�cientsWe have computed optimal multi-stage coe�cients based on preconditioned Eu-ler discretizations incorporating the modi�ed Roe ux. Though these coe�cientshave been computed based on this speci�c operator, it is expected that they willprovide good high-frequency damping with other discrete spatial operators for thepreconditioned Euler equations as well. (Tai's coe�cients [56], which are based ona one-dimensional scalar operator, have been used with success with di�erent spa-tial discretizations of the Euler equations; these coe�cients should work in a similarmanner).The functional for these optimizations was computed using the entire high-frequencydomain minus the points �ltered out to keep the problem properly posed (cf. Sec-tion 4.3 regarding the alignment and sonic problems). Since �x and �y take on acontinuum of values (though the number of modes that can actually be representedis mesh-dependent), the high-frequency domain was suitably discretized in order tosolve the problem numerically. The ampli�cation factor associated with the multi-stage scheme is a fairly smooth function of �x and �y and it is therefore easy tocompute the maximum value of the ampli�cation factor over the frequency domainin question.The coe�cients are presented in Tables 4.1 - 4.4. �opt is the maximum value ofthe ampli�cation factor in the high-frequency domain (minus wedge). jP jmax is themaximumvalue of the ampli�cation factor over the entire frequency range. M and �were taken as 0:1 and 0� respectively for these calculations. These tables have beenpresented earlier in [34]. Some of the coe�cients presented in this reference haveactually been improved upon, a result of the more robust optimization algorithm(simulated annealing) that was used to obtain these coe�cients. Also, the length

90�7:50�6:00�4:50�3:00�1:50 0:00 1:50 3:00�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:5303Modi�ed Roe scheme with � = 0, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00113Figure 4.19: As Figure 4.12 but for � = 0.�7:50�6:00�4:50�3:00�1:50 0:00 1:50 3:00�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:4938Modi�ed Roe scheme with � = �1, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.004 11Figure 4.20: As Figure 4.12 but for � = �1.

91�5:00�4:00�3:00�2:00�1:00 0:00 1:00 2:00�3:50�2:50�1:50�0:500:501:502:503:50

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:5481Modi�ed Roe scheme with � = 1=3, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00Figure 4.21: As Figure 4.12 but for � = 1=3.�7:00�5:50�4:00�2:50�1:00 0:50 2:00 3:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:6641Modi�ed Roe Scheme with � = 1=3, M = 0:1, � = 45�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.010Figure 4.22: Sub-optimal 4-stage scheme obtained in [34]. M = 0:1, � = 45�. �1 =0:1624, �2 = 0:2755, �3 = 0:5025, � = 1:6073, �opt = 0:6641

92scale was presented di�erently in [34]. The optimal coe�cients presented in thisreference were based on footprints withM = 0:1 and � = 45. In order to be consistentwith 1-D Courant numbers as given by Tai and others, the Courant numbers in thetables in this reference need to be scaled up by a factor of p2.Some design graphs for higher-order (�-) discretizations are presented in Figures4.19 - 4.21. At �rst glance, the stability domains in these �gures still do not appearto have an optimal shape. Judging from the design-graphs for the �rst-order operator(Figure 4.12), we would expect a much snugger �t of the stability domain aroundthe footprint. For higher-order footprints, which have fourth-order concact with theimaginary axis, a snug �t apparently is not the proper criterion. Figure 4.22 showsa snug �t for � = 1=3 for the same case as in Figure 4.21. It is seen that �opt is 20%higher with the snug-�t solution. Coe�cients are presented in the legend to Figure4.22 for comparison; they di�er substantially from those in Table 4.21. If we were toplot the contour corresponding to �opt for Figure 4.21, we would �nd that the contourtouches the outline of the footprint at its low-frequency extremities, indicating thatthese points have equal values of jP j. This is what we would expect from an optimalscheme.These coe�cients are also close to the coe�cients obtained based on optimizingfor the 2-D convection spatial operator with convection angle 45�. For a ow angle of45�, the functional based on the preconditioned Euler footprint is governed primarilyby the entropy/shear eigenvalues and their corresponding ampli�cation factors. (At0�, the acoustic eigenvalues also play a role in determining the value of the functional).These coe�cients may therefore be considered to be an analog to the coe�cientsderived by Tai [66, 56] based on 1-D convection.Other than for the Courant numbers, the coe�cients are reasonably close to Tai's

93for the �rst-order discretization (within 5%). For example, Tai's �rst-order four-stagecoe�cients are given by (�1 = 0:0833, �2 = 0:2069, �3 = 0:4265, � = 2:0), whereasthe �rst-order four-stage coe�cients from Table 4.1 are given by (�1 = 0:0787, �2 =0:2004, �3 = 0:4241, � = 2:64). However, this variation is signi�cant. If we wereto use the same Courant number (i.e. 2.64) with Tai's coe�cients, the value of �we obtain is only 0:329 as compared to the optimal value of 0:235. Variations of30% are observed in the coe�cients for higher-order discretizations. For example,for � = 1=3, Tai's four-stage coe�cients are given by (�1 = 0:1666, �2 = 0:3027,�3 = 0:5275, � = 1:732) as compared to (�1 = 0:1806, �2 = 0:3627, �3 = 0:7152,� = 1:5369) in Table 4.4.With the exception of the 2- and 3-stage schemes for � = �1, all the otherschemes have jP jmax = 1, indicating that these schemes are stable. The 2- and 3-stage schemes for � = �1 exhibit mild instability, with jP j slightly greater than 1 forthe some low frequency modes. This is probably a result of the formulation itself.Since this problem also occurs for � = �1 with the optimization over the high-highfrequency domain (cf. Section 4.6), it is likely to be a combination of the � = �1footprint and the optimization formulation that is creating the problem. The � = �1footprint is the most elongated along the negative real axis among all the �-schemesand is yet squeezed against the imaginary axis as well. There is a con ict betweenmoving zeros further left and still including the imaginary axis for stability. Oneway to ensure that all modes are damped or are neutrally stable is by adding in aconstraint to the optimization problem. However this complicates the problem andmakes its solution more di�cult to obtain (cf. Section 5.2). The mild instability issmall enough in magnitude that it will be o�set by the damping mechanism presentin a complete multigrid operator to create problems in general. We therefore opted

94not to enforce a global stability constraint. Figure 4.30 shows the design graph ofan optimal two-stage scheme along with the Fourier footprint of the entire domain.The mild instability present does not show up on the scale of this �gure.We have also computed the value of �1=�opt for all the tables presented. This valuenormalizes the damping of the scheme per unit CFL number. Tables 4.1 - 4.4 indi-cate that the normalized damping values increase as the number of stages increase,though the rate of increase tapers o� in some cases. Our numerical experiments (for�rst-order and � = 0 discretizations) indicate that four- or �ve-stage schemes arelikely to provide the best convergence rates for multigrid with local preconditioning.This is contrary to Tai's [56] experience that 2- or 3-stage time-stepping schemesperform best in a multigrid setting. The coe�cients presented below are designedfor the preconditioned Euler spatial operator, unlike Tai's coe�cients. As long asthe normalized damping coe�cients do not decrease we would expect more stagesto provide more damping and better performance. In general, however, the numer-ical implementation of a multigrid method does not appear to require that all thehigh-frequency components be damped down to zero by the relaxation scheme. Ournumerical experiments indicate that a 6-stage scheme is likely to be more expensivein the long run even if it does provide better damping than a 5-stage scheme.It might be more appropriate to use �1=� (instead of �) as the functional to beoptimized. We do not think that this will make much of a di�erence in the sets ofcoe�cients, while the optimization process might become more troublesome becauseof the more complicated functional. It has also been observed that sub-optimalschemes designed based on � have higher values of � as well as �1=�.

95Number of Stages2 3 4 5 6�1 0.2873 0.1316 0.07876 0.05134 0.03571�2 1 0.3755 0.2004 0.1235 0.08313�3 1 0.4241 0.2359 0.1507�4 1 0.4475 0.2581�5 1 0.4623�6 1� 1.4426 2.0912 2.6364 3.1822 3.7477�opt 0.5067 0.3451 0.2349 0.1569 0.09480�1=�opt 0.6242 0.6012 0.5773 0.5588 0.5333jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 4.1: Optimal multi-stage coe�cients for �rst-order scheme. Optimizationbased on entire high-frequency domain minus �ltered region (see Section4.3).

96Number of Stages2 3 4 5 6�1 0.4978 0.2075 0.1382 0.09621 0.06722�2 1 0.5915 0.3079 0.2073 0.1438�3 1 0.6185 0.3549 0.2391�4 1 0.6223 0.3710�5 1 0.6119�6 1� 0.8799 1.1868 1.4655 1.7719 2.1140�opt 0.7819 0.6393 0.5303 0.4337 0.3505�1=�opt 0.7561 0.6859 0.6487 0.6241 0.6090jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 4.2: Optimal multi-stage coe�cients for � = 0 scheme. Optimization basedon entire high-frequency domain minus �ltered region (see Section 4.3).

97Number of Stages2 3 4 5 6�1 0.3474 0.1619 0.09710 0.06536 0.04722�2 1 0.4611 0.2435 0.1552 0.1086�3 1 0.5104 0.2903 0.1928�4 1 0.5339 0.3192�5 1 0.5456�6 1� 0.6527 0.9050 1.1432 1.3737 1.5993�opt 0.7572 0.6169 0.4938 0.3916 0.3096�1=�opt 0.6530 0.5864 0.5394 0.5054 0.4804jP jmax 1.0099 1.0013 1.0000 1.0000 1.0000Table 4.3: Optimal multi-stage coe�cients for � = �1 scheme. Optimization basedon entire high-frequency domain minus �ltered region (see Section 4.3).

98Number of Stages2 3 4 5 6�1 0.6621 0.2667 0.1806 0.1249 0.07663�2 1 0.7525 0.3627 0.2456 0.1627�3 1 0.7152 0.3916 0.2624�4 1 0.6767 0.3932�5 1 0.6565�6 1� 0.9335 1.2147 1.5369 1.9708 2.3403�opt 0.7810 0.6320 0.5481 0.4500 0.3695�1=�opt 0.7674 0.6853 0.6762 0.6669 0.6535jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 4.4: Optimal multi-stage coe�cients for � = 1=3 scheme. Optimization basedon entire high-frequency domain minus �ltered region (see Section 4.3).

990:0 15:0 30:0 45:00:7000:8501:0001:150

� (degrees)N=(N)0� Variation with ow angle1st order 4-stage scheme �1=(�1)0��2=(�2)0��3=(�3)0��1d=(�1d)0�1=(cos�+AR sin�)Figure 4.23: Variation with ow angle of multi-stage coe�cients and Courant number(based on a �xed length scale independent of ow angle) for a �rst-order4-stage optimal scheme. Optimization over high-high frequency domainonly.4.6 Optimization over High-High Frequency DomainIf we make the assumption that the multi-stage scheme is to be used in conjunc-tion with a semi-coarsening multigrid algorithm, then the multi-stage scheme has tobe a good damper of only the high-high frequency error-modes [2].By restricting our functional to the high-high frequency domain, we are able toavoid both the alignment problem and the singularity problem for M ! 1. Thealignment problem is related to high-low or low-high entropy/shear error-modes.Since we no longer have to include these modes in the optimization, the optimizationproblem is properly formulated even for cases when the ow is aligned with thegrid. Furthermore, the deterioration of the damping properties for M ! 1 (or forAR !1) does not manifest itself in the high-high frequencies, as shown in Section4.6.1.Any variation in the high-high frequency content of the footprint with ow an-

100�2:60�2:20�1:80�1:40�1:00�0:60�0:20 0:20�1:40�1:00�0:60�0:200:200:601:001:40

<(z)=(z) Modi�ed Roe Scheme with M = 0:1, � = 0�, � = 1.Fourier Footprint of hi-hi domain.Figure 4.24: Fourier footprint of the �rst-order upwind approximation of the spatialEuler operator with the preconditioner of Van Leer et al. for M = 0:1,and ow angle � = 0�. Footprint corresponds to high-high frequencydomain.gle � can accounted for by de�ning the length-scale used in the Courant number asbefore (Equation (4.11)). Figure 4.23 shows the variation in Courant number withlength scale independent of ow angle for a typical multi-stage scheme, as obtainedfrom actual optimization and from the formula. As can be seen, our choice of lengthscale provides a fairly good �t to the true variations. The theoretical value under-estimates the true value by at most 7%, which is safe regarding stability and doesnot raise � much (to 0.09 for this case), since the functional is at near the optimumcon�guration. This allows us to recommend a single value of � for each multi-stagescheme.Figure 4.24 is an example of the high-high frequency content in the Fourier foot-print of the modi�ed Roe scheme. Figure 4.25 is the design-graph of a multi-stagescheme that was obtained by optimizing over this high-high frequency domain. Notethat this scheme is a much better damper of the high-frequency modes in its op-

101�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Modi�ed Roe scheme, M = 0:1, � = 0�Optimal 4-stage scheme. 1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.0011Figure 4.25: Design-graph of optimal �rst-order upwind four-stage scheme obtainedby optimizing over its high-high frequency footprint. M = 0:1, � = 0.�opt = 0:0632.timization domain than the previous 4-stage scheme (Figure 4.12) for the entirehigh-frequency footprint: �opt drops from 0.2362 down to 0.0632.4.6.1 Dependence on Mach numberThe Mach-number dependence of the multi-stage schemes derived in the previoussection was related to high-low acoustic error modes that had symbols at some dis-tance to the origin vanishing as p1�M2. The high-high frequency error-modes areinsensitive to Mach number, making it possible to design multi-stage schemes withcoe�cients that are close to optimal regardless of the Mach number. Figure 4.26shows the variation in the optimal multi-stage coe�cients with Mach number. Aslight change in the coe�cients is observed at while passing through the sonic point,but the change in values is small enough to allow us to recommend a single set ofcoe�cients for any multistage scheme, applicable over the entire Mach-number and

1020:10 0:50 0:90 1:300:700:901:101:30

MN=(N)0:1 Variation with Mach number1st order 4-stage scheme �1=(�1)0:1�2=(�2)0:1�3=(�3)0:1�=(�)0:1Figure 4.26: Variation of multi-stage coe�cients and Courant number with Machnumber for a �rst-order 4-stage optimal scheme. Optimization overhigh-high frequency domain only. ow-angle range. (Likewise, variations with the aspect ratio are negligible). Thecoe�cients in the tables in the next section were selected based on M = 0:1 and� = 0� (and AR = 1) and yield almost optimal damping regardless of the values ofM and �.4.6.2 Optimal multi-stage coe�cientsAs before, we have computed optimal multi-stage coe�cients (Tables 4.5 - 4.8)based on preconditioned Euler discretizations incorporating the modi�ed Roe ux.These multi-stage coe�cients are again expected to provide good high-frequencydamping with other discrete spatial operators for the preconditioned Euler equationsas well. These coe�cients are also close to the coe�cients obtained by optimizingbased on the corresponding 2-D convection operator, though not as close as before,when the optimization involved the entire high-frequency domain. In optimizing overthe high-high frequency domain, the acoustic high-high Fourier modes play a larger

103�7:50�6:00�4:50�3:00�1:50 0:00 1:50 3:00�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:2047Modi�ed Roe scheme with � = 0, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00Figure 4.27: As Figure 4.25 but for � = 0.role; since these modes do not exist for a purely convective operator, di�erences inthe multi-stage coe�cients would be expected.Figures 4.27 to 4.29 are examples of design graphs obtained with the optimalmulti-stage coe�cients presented in this section.The functional � for these optimizations was computed based on the high-highfrequency domain of the footprint. As before, the frequency domain was suitablydiscretized in order to solve the problem numerically.With the exception of the 2-4 stage schemes for � = �1, all the schemes havejP jmax = 1, indicating that these schemes are stable. The 2-4 stage schemes for � =�1 exhibit mild instability, with jP j slightly greater than 1 for some low frequencymodes (see Figure 4.30). The instability is even weaker than for the unstable � =�1 schemes of Section 4.5 (optimization over entire high-frequency domain) and isunlikely to endanger the stability of the full multigrid operator.Though 2-stage coe�cients have been given for higher-order discretizations, it

104�7:50�6:00�4:50�3:00�1:50 0:00 1:50 3:00�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:2526Modi�ed Roe scheme with � = �1, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00Figure 4.28: As Figure 4.25 but for � = �1.�5:00�4:00�3:00�2:00�1:00 0:00 1:00 2:00�3:50�2:50�1:50�0:500:501:502:503:50

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:3171Modi�ed Roe scheme with � = 0, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00Figure 4.29: As Figure 4.25 but for � = 1=3.

105�4:50�3:50�2:50�1:50�0:50 0:50 1:50 2:50�3:50�2:50�1:50�0:500:501:502:503:50

<(z)=(z) Optimal 2-stage, �(~�; �) = �opt = 0:6154Modi�ed Roe scheme with � = �1, M = 0:1, � = 0�1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00Figure 4.30: Design graph of optimal two-stage scheme based on high-high frequencydomain together with Fourier footprint of entire spatial operator. � =�1 spatial discretization. There is a small transgression of the footprintacross the stability boundary near the origin (invisible here).is not recommended that they be used in multigrid applications. Our experienceindicates that the semi-coarsenedmultigridmethod exhibits poor (or no) convergencewith 2-stage time-stepping and higher-order spatial discretizations. As in Section4.5, values of �1=�opt typically show improved damping per unit CFL number withmore stages in the time-stepping scheme, though the rate of increase in normalizeddamping values tapers o�. We believe that four- or �ve-stage schemes are likely toprovide the best convergence rates for multigrid with local preconditioning. Againno optimization with functional �1=� was attempted.The tables for �rst-order upwind and � = 0 discretizations have been presentedin [35]. It was found that the six-stage schemes presented there could be improvedupon slightly (even the simulated-annealing algorithm sometimes comes up withsub-optimal solutions). All the other stage-coe�cients are the same.Intuitively, it would again appear that the coe�cients used to generate Figure

1064.29 are not optimal. However, as mentionedwith regard to Figure 4.21 (from Section4.5), if we were to plot the contour corresponding to �opt for this �gure, we would �ndthat the contour touches the outline of the footprint at its low-frequency extremities,indicating that these points have equal values of jP j. This is what we would expectfrom an optimal scheme.We will consider some extensions to the ideas presented here in the next chapter.Numerical applications of some of the above multi-stage coe�cients will be presentedin Chapter VI.

107Number of Stages2 3 4 5 6�1 0.3333 0.1467 0.08125 0.05204 0.03712�2 1 0.3979 0.2033 0.1240 0.08516�3 1 0.4226 0.2343 0.1521�4 1 0.4381 0.2566�5 1 0.4525�6 1� 1.0000 1.5252 2.1058 2.6824 3.0827�opt 0.3333 0.1418 0.06328 0.03024 0.01627�1=�opt 0.3333 0.2779 0.2696 0.2714 0.2628jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 4.5: Optimal multi-stage coe�cients for �rst-order scheme. Optimizationbased on high-high frequency domain.

108Number of Stages2 3 4 5 6�1 0.5713 0.2239 0.1299 0.08699 0.06134�2 1 0.5653 0.2940 0.1892 0.1322�3 1 0.5604 0.3263 0.2201�4 1 0.5558 0.3425�5 1 0.5531�6 1� 0.6305 1.0458 1.4008 1.7471 2.0701�opt 0.6475 0.4279 0.2927 0.2047 0.1464�1=�opt 0.5019 0.4441 0.4160 0.4034 0.3953jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 4.6: Optimal multi-stage coe�cients for � = 0 scheme. Optimization basedon high-high frequency domain.

109Number of Stages2 3 4 5 6�1 0.4450 0.1780 0.09900 0.06431 0.04540�2 1 0.4774 0.2434 0.1509 0.1044�3 1 0.4913 0.2783 0.1846�4 1 0.5004 0.3030�5 1 0.5116�6 1� 0.4386 0.7439 1.0139 1.2608 1.4613�opt 0.6154 0.3916 0.2526 0.1654 0.1145�1=�opt 0.3306 0.2836 0.2574 0.2400 0.2269jP jmax 1.0002 1.0002 1.0001 1.0000 1.0000Table 4.7: Optimal multi-stage coe�cients for � = �1 scheme. Optimization basedon high-high frequency domain.

110Number of Stages2 3 4 5 6�1 0.6826 0.2695 0.1572 0.1085 0.07727�2 1 0.6351 0.3350 0.2180 0.1537�3 1 0.6079 0.3581 0.2428�4 1 0.5887 0.3650�5 1 0.5729�6 1� 0.7167 1.1704 1.5694 1.9924 2.4407�opt 0.6677 0.4454 0.3171 0.2336 0.1686�1=�opt 0.5692 0.5010 0.4810 0.4715 0.4822jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 4.8: Optimal multi-stage coe�cients for � = 1=3 scheme. Optimization basedon high-high frequency domain.

CHAPTER VFURTHER ANALYSISIn this chapter, we will expand upon the ideas presented in Chapter IV. In partic-ular we will consider the generalization to arbitrary quadrilateral cells, multi-stageschemes based on Navier-Stokes spatial operators, multi-stage schemes based ondiscretizations incorporating explicit residual-smoothing, and the extensions to un-structured meshes and three-space dimensions.5.1 De�ning the Courant Number Length-Scale for an Ar-bitrary Quadrilateral CellThe de�nition given earlier for the length scale l used in de�ning the Courantnumber (Section 4.4) requires the assumption that the cell is rectangular. It ispossible to extend this de�nition to a general quadrilateral cell.Consider the quadrilateral cell shown in Figure 5.1, with vertices �x1 to �x4. Thiscell �ts into a rectangular cell aligned with the ow direction; it has dimensions �x1and �y1 and aspect-ratio ARq = �x1=�y1.We will �rst obtain de�nitions for �x1 and �y1 in terms of the cell vertices �xk. Ifeq is the unit-vector in the direction of the ow and en the unit-vector in the direction111

112�x1

�x2�x3�x4 eq�x1

e2e1en

�y1Figure 5.1: Quadrilateral cell with equivalent rotated rectangular cell in the owdirection. The cell has an equivalent aspect-ratio ARq = �x1=�y1.normal to the ow, we have�x1 = maxk (�xk � eq)�mink (�xk � eq); (5:1)�y1 = maxk (�xk � en)�mink (�xk � en): (5:2)The value of ARq is used to compute the acoustic scale factor � according toEquation (3.35). The length scale l then follows froml = cell area�y1 ; (5:3)a generalization of Equation (4.12).Our numerical studies indicate that computing the length-scale for general quadri-lateral cells in this manner improves multigrid convergence rates by a factor of 30%or more compared to using a length scale independent of �, e.g. pcell area. The

113extra work required in computing this length scale can therefore be considered to beworth the e�ort.5.2 Multi-Stage Coe�cients for Navier-Stokes Spatial Op-eratorsIn designing Navier-Stokes preconditioners we use the idea put forth in [17] ofmaking the size of the footprint independent of cell-Reynolds number. In this waywe can generate a family of multi-stage schemes that are dependent only on cell-Reynolds number, and only if we wish so. If we write the 2-D discretized Navier-Stokes equations as:Ut = LEuU+ (CUx)x + (D1Ux)y + (D2Uy)x + (EUy)y; (5:4)the �rst term on the right-hand side is the discrete Euler operator; the remainingterms are the viscous/conductive terms, assumed to be approximated by centraldi�erencing. These contribute only to the extent of the footprint along the negativereal axis, which is inversely proportional to the cell-Reynolds number. The properscaling required to make the size of the footprint independent of cell-Reynolds numberis obtained by choosing PNS as follows:P�1NS = P�1Eu + 2�xC+ 2�yE: (5:5)For higher-order upwind di�erencing the same scaling technique for the highest-frequency Fourier modes yields a similar expression:P�1NS = (1 � �)P�1Eu + 2�xC + 2�yE: (5:6)This strategy works for cell-Reynolds numbers that are not too low, e.g. Re�x � 0:1.For very low Re�x a slight modi�cation is needed in the continuity equation; one

114suggestion [70] has been to add an element of order � 1=Re�x on the main diagonal.This will be discussed elsewhere (cf. [31]).After the scaling of the footprint, we could satisfy ourselves with the multi-stagecoe�cients developed for the Euler family, i.e., for Re�x ! 1. We would hope,though, that as the cell-Reynolds number decreases and the high-frequency con-tent in the footprint begins to align itself along the negative real axis, the timesteps required for good damping would automatically increase. This, unfortunately,does not happen if we apply the above optimization procedure to a discrete, pre-conditioned Navier-Stokes operator with prescribed values of Mach number, owangle, cell aspect-ratio and cell-Reynolds number. The formulation described in Sec-tion 4.3 simply attempts to maximize the damping over the domain, resulting inextremely, unnecessarily, small functional values (�opt) for the corresponding multi-stage schemes. This damping comes at a price - unnecessarily small Courant num-bers for these time-stepping schemes. This immediately leads to an increase in thenumber of computational steps required to attain a converged solution, whether bysingle-grid or multigrid relaxation. Since the cell-length scales used in Navier-Stokescalculations can be locally much smaller than those used in Euler calculations, thelarger the Courant number of the scheme, the better. It is therefore obvious that adi�erent approach is desirable for Navier-Stokes operators.5.2.1 Reformulating the optimization problemFor Navier-Stokes operators, we have rede�ned the optimization problem as fol-lows: For a given spatial operator, �nd the largest Courant number with which thecorresponding multi-stage scheme has a prescribed damping capability.

115This may be written as: Solve �(�) = �min (5:7)where �(�) is de�ned below:�(�) = min~� 0@ maxj�xj;j�yj2[�2 ;�] kP (z(�x; �y; �); ~�)k1A :The solution to this problem is fairly simple. We make the assumption that �(�)is a continuous, monotonically-increasing function within our range of interest. Thisseems to be a valid assumption, as borne by our experiments. We may then searchfor a root to Equation (5.7) using an algorithm such as the bisection method (cf.[46]) or the more complex algorithm due to Brent [8] which combines the surenessof the bisection method with the speed of a higher-order method when appropriate.Each evaluation of �(�) requires a solution to the underlying optimization problem(Section 4.3) which may be solved using simulated annealing as before (cf. AppendixA). Care must therefore be taken to ensure that an appropriate set of starting valuesof ~� are used for each evaluation of �(�). Of greater concern is the choice of �min.Too large a value of �min can result in a scheme that is grossly unstable for frequencymodes other than the high-high combinations being considered. Figure 5.4 illustratessuch an unstable scheme.Figure 5.2 is an example of the design-graph of a scheme successfully designedusing this procedure. In contrast, Figure 5.3 shows the design graph of a schemedesigned with the optimization procedure described earlier. As can be seen from the�gures, the two schemes have very di�erent damping properties. The optimizationwith oating � achieves a value of �opt as low as 0.0034, at a Courant numberof 2.95. By prescribing the damping factor �min to be 0.1, the scheme's Courantnumber increases to 7.89.

116�20:0�17:0�14:0�11:0 �8:0 �5:0 �2:0 1:0�7:0�5:0�3:0�1:01:03:05:07:0 <(z)=(z) First order preconditioned N-S operator.4-stage scheme with prescribed damping. 1 0.022 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.002 5 11

Figure 5.2: Four-stage scheme obtained by optimizing over high-high frequency foot-print of the discrete, preconditioned Navier-Stokes operator. �min wasprescribed as 0.1. M = 0:1, � = 0, Re�x = 0:1, AR = 1. � = 7.891.This approach might also be tried for discretizations of the Euler operator, butis not well suited for it. The increase in Courant number is marginal at best, forthe following reason. Navier-Stokes footprints tend to align themselves along thereal axis in the complex plane with decreasing cell-Reynolds number, as the behaviorbecomes more parabolic. Increasing the Courant number causes the footprint to growparticularly in this direction for these operators; the spread in the other dimensionremains unimportant. Euler footprints are \bulky" and grow outward in all directionsat the same rate, making it harder to stay within the stability domain.One way of avoiding multi-stage coe�cients that amplify some error-modes, as inFigure 5.4, is by adding in a contraint to the formulation. An appropriate constraintwould be Pmax � maxj�xj;j�yj2[0;�] kP (z(�x; �y; �); ~�)k � 1: (5:8)This constrained-optimization problem (Equations (5.7) and (5.8)) can be re-

117�13:0�11:0 �9:0 �7:0 �5:0 �3:0 �1:0 1:0�7:0�5:0�3:0�1:01:03:05:07:0 <(z)=(z) Preconditioned Navier-Stokes operator.Optimal 4-stage scheme. 1 0.022 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.0011

Figure 5.3: Optimal four-stage scheme obtained by optimizing over high-high fre-quency footprint of the discrete, preconditioned Navier-Stokes operator.M = 0:1, � = 0, Re�x = 0:1, AR = 1. �opt = 0.0034, � = 2.953.�28:0�24:0�20:0�16:0�12:0 �8:0 �4:0 0:0�7:0�5:0�3:0�1:01:03:05:07:0 <(z)=(z) First order preconditioned N-S operator.4-stage scheme with prescribed damping. 1 0.022 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.0011 11

Figure 5.4: Four-stage scheme obtained by optimizing over high-high frequency foot-print of the discrete, preconditioned Navier-Stokes operator. �min wasprescribed as 0.2. M = 0:1, � = 0, Re�x = 0:1, AR = 1. � = 12.845.This scheme will be unstable for some \high-low", \low-high" and \low-low" frequency combinations.

118formulated as an unconstrained-optimization problem by making use of a penaltyfunction. We can write �(�) as�(�) = min~� 0@ maxj�xj;j�yj2[�2 ;�] kP (z(�x; �y; �); ~�)k+ (Pmax � 1)21A ; (5:9)where is a constant, � 0. The larger is, the more strongly is the constraintenforced. Taking too large a value of however can a�ect the robustness of theoptimization algorithm. We found that taking taking a modest value of initially(around 100) and then recomputing the solution with this trial solution and around1:�106 worked well in enforcing the constraint. Note that this procedure would alsoremove the slight instability for the schemes represented in Tables 4.3 and 4.7. Figure5.5 is the design-graph of a solution generated in this manner. We have obtained astable scheme with only a modest decrease in Courant number (from 12.8 to 11.4). Itis unclear however if this schemewill perform well in nonlinear implementations, sinceit has some neutrally stable medium-frequencymodes. These are likely to be excitedif the time-step taken corresponds to even a modest increase over the prescribedvalue. One option is to modify our constraint so as to allow for an increase in theCourant number without losing stability.This modi�ed constraint can be written asP �max(�) � max�=�;�(1+�) maxj�xj;j�yj2[0;�] kP (z(�x; �y; �); ~�)k! � 1: (5:10)The optimization problemmay be formulated as an unconstrained optimization prob-lem as before. Figure 5.6 shows the solution to this problem (Equations (5.7) and(5.10)) with � = 0:2. The resulting coe�cients are likely to be more reliable in amultigrid formulation than the coe�cients derived earlier. Inspection of the designgraph makes one wonder, though, if this solution cannot still be improved upon.Choosing a moderate value for �min, say around 0.12 for the case considered

119�25:0�21:0�17:0�13:0 �9:0 �5:0 �1:0 3:0�7:0�5:0�3:0�1:01:03:05:07:0 <(z)=(z) First-order preconditioned N-S operator.4-stage scheme with prescribed damping. 1 0.022 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00113

Figure 5.5: Four-stage scheme obtained by optimizing over high-high frequency foot-print of the discrete, preconditioned Navier-Stokes operator with a con-straint on Pmax introduced via a penalty function. �min was prescribedas 0.2. M = 0:1, � = 0, Re�x = 0:1, AR = 1. � = 11.3679.�25:0�21:0�17:0�13:0 �9:0 �5:0 �1:0 3:0�7:0�5:0�3:0�1:01:03:05:07:0 <(z)=(z) First-order preconditioned N-S operator.4-stage scheme with prescribed damping. 1 0.022 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00

Figure 5.6: Four-stage scheme obtained by optimizing over high-high frequency foot-print of the discrete, preconditioned Navier-Stokes operator with a con-straint on P �max (Equation 5.10) introduced via a penalty function. �minwas prescribed as 0.2. M = 0:1, � = 0, Re�x = 0:1, AR = 1. � = 0:2. �= 9.5361.

120above, would obviate the need for the constraint and its associated complications.It is unclear what is the best choice for �min as regards obtaining the best possiblemultigrid convergence rates. Multi-stage schemes with �opt = 0:26 and more (notprescribed) have worked well in our multigrid Euler studies. It is not likely that theoptimal Euler values of � would be appropriate for use in Equation (5.7) for verylow cell-Reynolds numbers; the value of � would become too large to be considereda perturbation. One alternative would be to abandon Equation (5.7) and formulateand solve a more complex constrained optimization problem; we have not endeavoredthis.5.2.2 Multi-stage coe�cients with prescribed damping for Navier-StokesoperatorsIt is unlikely that we would be able to recommend a single set of coe�cientsindependent of M , Re, � and AR. A likely compromise would be to generate curve-�ts to account for the variations in coe�cients with some of these parameters.As a preliminary study we have obtained the variation in coe�cients with Re�x,for a 4-stage scheme based on a �rst-order discretization of the preconditioned Navier-Stokes operator. This is not a discretization one would use in practice, since thearti�cial viscosity would exceed the real viscosity on all except the �nest grids. Figure5.7 shows these variations, normalized to the values of the coe�cients for the Euleroperator. The �gure indicates that the optimal Courant number increases sharplyas the cell-Reynolds number decreases, while the multi-stage coe�cients decrease. Itwould seem that curve-�ts in terms of log(Re�x) are feasible.

121�2:0 1:0 4:0 7:00:001:503:004:50

log(Re�x )N=(N)Eu Variation with Cell Reynolds no1st order 4-stage scheme �1=(�1)Eu�2=(�2)Eu�3=(�3)Eu�=(�Eu)Figure 5.7: Variation with cell-Reynolds number of multistage coe�cients andCourant number for a �rst-order 4-stage scheme with prescribed damp-ing. Optimization based high-high frequency footprint of the discrete,preconditioned Navier-Stokes operator with a constraint on Pmax intro-duced via a penalty function. �min was prescribed as 0.15. M = 0:1,� = 0, AR = 1, � = 0:05.5.3 Discretizations Incorporating Explicit Residual Smooth-ingStability restrictions place a limit on the maximum time-step that may be takenwhile marching to the steady state. The Courant numbers associated with explicittime-stepping stepping schemes are relatively small when compared with those ofimplicit methods. This slows down the process of eliminating errors by convectionout of the domain.It was observed by Jameson [24] that residual smoothing helps alleviate thisrestriction to some extent. Residual smoothing, or residual averaging, involves theuse of weighted averages of the residuals from neighboring cells when computing atemporal update. More speci�cally, once the residual Rij has been computed in theijth cell, an additional explicit step can be added in order to average the residual

122variations. This increases the support of the scheme and damps certain error-modes,so that larger time-steps can be taken in the update scheme. The averaging can bewritten as �Rij = �1 + �(�2x + �2y)�Rij; (5:11)where � represents a central-di�erencing operator and � a smoothing parameter.This explicit-smoothing step is extremely cheap to compute in comparison to ux-function computations. One primary disadvantage of explicit residual smoothing isa severe stability restriction on � and, therefore, on the smoothing e�ect. For thisreason implicit residual smoothing is preferred in implementations. Implicit residualsmoothing is more expensive however.Residual smoothing can modify the Fourier footprint in a bene�cial manner,resulting in the ability to design optimal multi-stage schemes with larger Courantnumbers. Tai et al. [57], again using 1-D analysis, have derived optimal schemes foruse with implicit residual smoothing. We will focus on explicit residual-smoothingin our study. The low cost of this smoothing step makes it an extremely attractiveoption, restriction on � notwithstanding.5.3.1 Choice of smoothing stencilOur analysis of various stencils on a square mesh, vis. standard �ve-point Lapla-cian, rotated �ve-point Laplacian, standard 9-point Laplacian, a combination ofstandard and rotated 5-point Laplacians, indicates that the standard 5-point Lapla-cian stencil for smoothing provides the best footprint for designing optimally-dampedschemes. (The 9-point stencil was discarded because of its severe restriction on �:� < 1=32).If we insert Fourier data into Equation (5.11) we obtain, for the standard 5-point

123stencil, FT ( �Rij) = (2�(cos �x + cos �y) + (1 � 4�))FT (Rij):It is necessary to ensure that the ampli�cation factor of the averaging operatorhas non-zero values. If this condition is not met, �Rij could be zero when Rij is not,which would disable the update scheme. We therefore require that the ampli�cationfactor remain positive; using the condition�2 � cos �x + cos�y � 2yields � < 18 : (5:12)The rotated �ve-point stencil has the same bound on � (i.e. Equation 5.12). How-ever as Figures 5.8 and 5.9 indicate, the rotated stencil generates \bulky" footprintsthat prohibit the formulation of schemes with good damping properties as well aslarge Courant numbers. The standard stencil generates footprints that \fold in" thehigh frequencies towards the origin.5.3.2 Optimal multi-stage coe�cientsFigure 5.10 is the design-graph of an optimal scheme based on a footprint incor-porating residual smoothing (�xed � = 0:1). A comparison with Figure 4.10 showsthat the Courant number increases from 1.86 to 3.25.It is possible to incorporate � in the optimization so as to obtain a value of � thatincreases the overall damping of the high-frequency error-modes, i.e. that decreases�opt. However, in most cases the solution with � = 0, i.e. no smoothing, has thestrongest overall damping (lowest value of �opt), so the optimization would movetowards this solution. Our experiments indicate that � = 0:1 is a good choice when

124�2:50�2:00�1:50�1:00 �:50 :00 :50 1:00�1:75�1:25�:75�:25:25:751:251:75

<(z)=(z) Fourier Footprint.1st order with RSFigure 5.8: Fourier footprint of the �rst-order preconditioned Euler operator with ex-plicit residual smoothing. Regular Laplacian stencil used for smoothing.M = 0:1, � = 45�, � = 0:1.

�3:50�3:00�2:50�2:00�1:50�1:00 �:50 :00�1:75�1:25�:75�:25:25:751:251:75<(z)=(z) Fourier Footprint.1st order with RS

Figure 5.9: Fourier footprint of the �rst-order preconditioned Euler operator with ex-plicit residual smoothing. Rotated Laplacian stencil used for smoothing.M = 0:1, � = 45�, � = 0:1.

125�9:00�7:50�6:00�4:50�3:00�1:50 :00 1:50�5:25�3:75�2:25�:75:752:253:755:25

<(z)=(z) Preconditioned Euler operator with RS.Optimal 4-stage scheme. 1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.0011Figure 5.10: Four-stage scheme obtained by optimizing over footprint of the �rst-order preconditioned Euler operator with explicit residual smoothing.Regular Laplacian stencil used for smoothing. M = 0:1, � = 45�,� = 0:1, � = 3:25.considering the high-frequency domain. In consequence, we did not incorporate �into the optimization.We have derived optimal multi-stage coe�cients for use with regular multigridand explicit residual smoothing (Tables 5.1 - 5.4) based on the high-frequency contentof the footprint for the 2-D preconditioned discrete Euler operators including themodi�ed Roe ux. For coe�cients that may be used with implicit residual smoothing,we refer to [57]. (Remember, though, that the coe�cients in that reference arederived based on a 1-D convection operator). As can be seen from comparing thetables with those in Chapter IV, the gain in Courant number is fairly substantialfor all schemes. Our experiments indicate that the factor de�ned earlier in orderto account for ow-angle variations (Equation (4.11)) is also well suited for thesemulti-stage schemes with residual smoothing.As before (in Section 4.5), �opt is the maximum value of the ampli�cation factor

126in the high-frequency domain (minus wedge). jP jmax is the maximum value of theampli�cation factor over the entire frequency range. M and � were taken as 0:1 and0� respectively, for these calculations. The value of the smoothing parameter � wastaken to be 0:1. The value of jP jmax is again slightly greater than 1 for some of thecoe�cients derived for � = �1.Tables 5.5 - 5.8 give coe�cients for use with semi-coarsening. The coe�cientsbased on the high-high frequency content of the Fourier footprint were obtainedwith a constraint on the optimization. It was again necessary to apply the con-straint jP jmax � 1 in order to obtain stable time-stepping schemes. This constrainedproblem was solved as in as in Section 5.2 (cf. Equations (5.8), (5.9)). The value ofjP jmax is therefore not presented in these tables.Interestingly, other than for � = �1, the values of �1=�opt are almost constant forthese schemes, whether optimized over all high or just the high-high frequencies. Wehave not performed enough multigrid experiments to be able to suggest how manystages are required for best performance in a multigrid strategy.As before, these schemes are not only preferable as solvers in a multi-grid strategy,but are also superior single-grid schemes. The residual-smoothing operator yields asubstantial gain in the allowable time-step while the additional computational workrequired is negligible in comparison to a ux computation.5.4 Optimal Schemes for Three-Dimensional Euler Opera-torsThe footprints of discretizations of three-dimensional spatial Euler operators showthe e�ect of having three di�erent frequency variables. Most prominent is that theacoustic part of the footprint, after preconditioning, has three \holes" rather than

127Number of Stages2 3 4 5 6�1 0.3438 0.1560 0.08541 0.05263 0.02628�2 1 0.4031 0.2027 0.1215 0.07019�3 1 0.4215 0.2284 0.1339�4 1 0.4334 0.2365�5 1 0.4377�6 1� 1.8191 2.6538 3.6334 4.6686 5.0753�opt 0.4554 0.3174 0.2071 0.1312 0.1115�1=�opt 0.6489 0.6489 0.6483 0.6472 0.6491jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 5.1: Optimal multi-stage coe�cients for �rst-order scheme with explicit resid-ual smoothing, � = 0:1. Optimization based on entire high-frequencydomain minus �ltered region (see Section 4.3).

128Number of Stages2 3 4 5 6�1 0.6007 0.2726 0.1688 0.1065 0.08262�2 1 0.6347 0.3231 0.2003 0.1504�3 1 0.6053 0.3324 0.2398�4 1 0.5744 0.3629�5 1 0.5676�6 1� 1.1082 1.5377 2.0397 2.6681 3.1243�opt 0.7326 0.5848 0.4954 0.3989 0.3340�1=�opt 0.7552 0.7055 0.7087 0.7087 0.7040jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 5.2: Optimal multi-stage coe�cients for � = 0 scheme with explicit residualsmoothing, � = 0:1. Optimization based on entire high-frequency domainminus �ltered region (see Section 4.3).

129Number of Stages2 3 4 5 6�1 0.3974 0.1770 0.1123 0.07707 0.05396�2 1 0.4931 0.2627 0.1743 0.1187�3 1 0.5318 0.3097 0.2033�4 1 0.5500 0.3266�5 1 0.5498�6 1� 0.9095 1.2300 1.5360 1.8377 2.1808�opt 0.7109 0.5598 0.4396 0.3437 0.2615�1=�opt 0.6872 0.6239 0.5856 0.5592 0.5406jP jmax 1.0138 1.0001 1.0000 1.0000 1.0000Table 5.3: Optimal multi-stage coe�cients for � = �1 scheme with explicit residualsmoothing, � = 0:1. Optimization based on entire high-frequency domainminus �ltered region (see Section 4.3).

130Number of Stages2 3 4 5 6�1 0.7216 0.3744 0.2091 0.1337 0.08905�2 1 0.6196 0.3093 0.2035 0.1472�3 1 0.6051 0.3413 0.2308�4 1 0.5671 0.3482�5 1 0.5532�6 1� 1.1537 1.7765 2.5115 3.3211 3.9865�opt 0.7606 0.6407 0.5533 0.4461 0.3794�1=�opt 0.7888 0.7783 0.7901 0.7842 0.7842jP jmax 1.0000 1.0000 1.0000 1.0000 1.0000Table 5.4: Optimal multi-stage coe�cients for � = 1=3 scheme with explicit residualsmoothing, � = 0:1. Optimization based on entire high-frequency domainminus �ltered region (see Section 4.3).

131Number of Stages2 3 4 5 6�1 0.3349 0.1534 0.08650 0.05647 0.03818�2 1 0.3933 0.2032 0.1265 0.08525�3 1 0.4190 0.2348 0.1501�4 1 0.4350 0.2521�5 1 0.4449�6 1� 2.2467 3.4119 4.6569 5.7945 6.9393�opt 0.3718 0.2144 0.1294 0.07161 0.04147�1=�opt 0.6438 0.6368 0.6446 0.6344 0.6321Table 5.5: Optimal multi-stage coe�cients for �rst-order scheme with explicit resid-ual smoothing, � = 0:1. Constrained optimization based on high-highfrequency domain.

132Number of Stages2 3 4 5 6�1 0.6515 0.2906 0.1739 0.1151 0.04587�2 1 0.5971 0.2961 0.1869 0.1309�3 1 0.5737 0.3367 0.2036�4 1 0.5477 0.3349�5 1 0.5356�6 1� 1.2329 1.7958 2.5268 3.0737 3.5383�opt 0.6653 0.5254 0.4307 0.3262 0.2682�1=�opt 0.7185 0.6988 0.7165 0.6946 0.6894Table 5.6: Optimal multi-stage coe�cients for � = 0 scheme with explicit residualsmoothing, � = 0:1. Constrained optimization based on high-high fre-quency domain.

133Number of Stages2 3 4 5 6�1 0.4961 0.1875 0.1121 0.07622 0.04704�2 1 0.5639 0.2616 0.1690 0.1107�3 1 0.5325 0.3028 0.1939�4 1 0.5298 0.3150�5 1 0.5249�6 1� 0.8411 1.2734 1.8678 2.3181 2.7183�opt 0.6679 0.4547 0.3078 0.2208 0.1696�1=�opt 0.6189 0.5385 0.5321 0.5212 0.5206Table 5.7: Optimal multi-stage coe�cients for � = �1 scheme with explicit resid-ual smoothing, � = 0:1. Constrained optimization based on high-highfrequency domain.

134Number of Stages2 3 4 5 6�1 0.6998 0.3537 0.2162 0.1512 0.1126�2 1 0.5897 0.3040 0.1964 0.1550�3 1 0.5981 0.3497 0.2535�4 1 0.5461 0.3639�5 1 0.5440�6 1� 1.2540 1.9857 2.7667 3.7120 4.1585�opt 0.7439 0.6050 0.5264 0.3950 0.3546�1=�opt 0.7898 0.7764 0.7930 0.7786 0.7793Table 5.8: Optimal multi-stage coe�cients for � = 1=3 scheme with explicit resid-ual smoothing, � = 0:1. Constrained optimization based on high-highfrequency domain.

135the two appearing in two dimensions (cf. Figure 3.5). As a result of this tripartition,high frequencies are found somewhat closer to the origin than in the correspondingtwo-dimensional footprint. This puts stronger demands on the multi-stage schemeif good high-frequency damping is to be achieved. We expect that six stages, ratherthan �ve or four would be needed to achieve the same quality of damping as in twodimensions.The extra demands on the multi-stage scheme disappear completely when op-timization of damping is restricted to the high-high-high frequency combinations,as allowed when combined with semi-coarsened multigrid relaxation. The problemhere is that semi-coarsened multigrid has not yet been fully developed for three-dimensional PDEs. If all semi-coarsened grids were to be included in the cycle, thetotal work would grow unboundedly as more and more grid levels were introduced(for problems with �ner and �ner base grids). Current research points in the direc-tion of using \sparse grids", i.e., using a subset of semi-coarsened grids on which allerror components can be represented [20].5.5 Discretizations on Unstructured MeshesUnstructured grids play an important role in the simulation of complex owsabout complicated geometries. It is therefore necessary to consider whether the kindof optimal schemes described in the previous chapter are applicable to unstructuredmeshes and their associated numerical algorithms. Deriving multi-stage coe�cientswith optimal local-damping properties is complicated by the fact that it is generallydi�cult, if not impossible, to apply Fourier analysis to numerical algorithms onthese meshes. As a result, the Fourier footprints that were used in the optimizationformulations this far would no longer be obtainable.

136�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimal 4-stage, �(~�; �) = �opt = 0:3559k = 0 cell-vertex scheme, � = 30� 1 0.052 0.103 0.204 0.305 0.406 0.507 0.608 0.709 0.8010 0.9011 1.00Figure 5.11: Optimal four-stage scheme based on the Fourier footprint of the k =0 cell-vertex scheme for the 2-D convection equation on a triangularmesh. � = 30�, �1 = 0:07051, �2 = 0:1803, �3 = 0:3854, � = 3:1079,�opt = 0:3559.5.5.1 Discretizations on triangular meshesObtaining the Fourier footprint for the �rst-order upwind cell-centered �nite vol-ume scheme on the simplest possible mesh, viz. an equilateral triangular mesh, isalready complicated by the fact that the centroids of the triangles do not lie in astraight line. Fourier analysis requires the use of a complicated mapping in thiscase. Cell-vertex schemes would appear to o�er more promise. However, most of therelatively new family of \ uctuation-splitting" cell-vertex schemes (cf. [14, 38]) arenonlinear, with the linear LDA scheme rarely used being in practice.Darmofal [12] has derived the Fourier footprint for Barth's k = 0 (�rst-order)upwind scheme [4] for the 2-D convection equation on a triangular mesh. Assumingthat there are six equilateral triangles meeting in a node, it is possible to de�neFourier variables �1, �2 and �3 in the directions of the edges radiating from the

137interior node. (Only two of these variables are actually signi�cant. The third variableis related to the other two through a compatibility relation). Figure 5.11 is the design-graph of an optimal scheme based on this footprint. It is interesting to note thatthis discretization does not su�er from alignment.The high-frequency part of the footprint used in the optimization contains allmodes for which at least one of the three spatial frequencies is in the high-frequencyrange. The convection angle is at 30� to the x-axis; one set of edges is alignedwith this axis. The footprint then has the largest extent; the footprints for allother convection angles appear to be contained in it. This case could therefore beconsidered to be the worst case for the design of multi-stage coe�cients based onthis spatial discretization.These multi-stage coe�cients obtained above have been been tested by Darmofal[12, 13] in a single-grid scheme on unstructured triangular meshes and appear tohave good damping properties. The coe�cients are �1 = 0:07051, �2 = 0:1803,�3 = 0:3854 and �4 = 1:0 and are close to those obtained in Section 4.5 for afour-stage scheme based on a �rst-order discretization of the Euler operator. TheCourant numbers are scaled di�erently and would have to be normalized in orderto be compared. It is therefore our belief that the coe�cients derived earlier, whenused with discretizations on unstructured triangular grids, would achieve dampingproperties reasonably close to those for the truly optimal coe�cients, provided thelength-scale in the Courant number is de�ned appropriately. The proper choice ofthe length-scale is critical in achieving good damping, and hence good multigridperformance; we have not attempted to derive such a scale here.

1385.5.2 Discretizations on Cartesian-unstructured meshesCartesian-unstructured meshes created by local re�nement have gained in popu-larity over recent years due to their relative simplicity and ability to resolve complex ows and geometries relatively cheaply. Fourier analyis of spatial discretizations onthese meshes becomes impossible as a result of the multiplicity of grid scales thatare present on a single grid level. It is the transition from one grid scale to the nextthat creates problems when attempting to use Fourier data and it is not clear whathappens at these interfaces in terms of local damping of error-modes.One way of studying the damping properties of a multi-stage scheme on thesemeshes is by taking a more global view of damping. We can the estimate the mag-nitude of the high-frequency damping for the update operator for one multi-stageupdate if the operator is linear and when the initial data are periodic. This valuecould be used as an indicator of the damping properties of the multi-stage scheme.The problem with this approach lies with the fact that this value is likely to bedependent on the particular mesh con�guration. It is also di�cult to analyze howwell error-modes are being damped locally, for example at an interface where the cell-lengths change. Finally, the information obtained is insu�cient for an optimizationof the multi-stage coe�cients with regard to high-frequency damping.On the positive side, De Zeeuw [72] has successfully used Tai's multi-stage co-e�cients [56] in multigrid solutions on Cartesian-unstructured meshes. We wouldtherefore expect the coe�cients derived earlier (e.g. in Section 4.5) to work as well,or even better, on these meshes for multigrid with local preconditioning, as they aree�ectively the 2-dimensional analog to Tai's coe�cients.We will now consider some numerical test cases.

CHAPTER VINUMERICAL STUDIESThis chapter will focus on numerical experiments in an attempt to validate theideas considered this far. We will �rst consider the propagation and decay of initiallyrandom perturbations on a square mesh. Studying this essentially linear case isjusti�ed on the grounds that it provides an insight into how much convergence ratescan be improved by using characteristic time-stepping with and without multigrid.Similar single-grid studies have been conducted by Lee [33] for the convection of apoint disturbance on a square domain.We will then apply characteristic time-stepping together with multigrid to thesimple nonlinear problem of \ ow past a semi-circular bump in a channel" in orderto highlight the improvements allowed by our approach in the nonlinear regime.All our numerical studies that include preconditioning utilize the preconditioner ofVan Leer et al. together with the scaling modi�cation described by Lee and Van Leer[32]; see Section 3.9. We will also consider the e�ects of explicit residual-smoothingon convergence.Our basic measure of work is the \work-unit", which is de�ned to be the amountof work required to compute a single-stage update on the �nest grid. As an exam-ple, a single �ne-grid update with a �ve-stage time-stepping scheme requires work139

140equivalent to �ve work-units (5 WU). Our estimates of work-units for multigrid cy-cles include all ux computations performed, including those required to computethe residual-restriction operators in the FAS method. Other operations, includingresidual smoothing, contribute insigni�cantly to the overall computational cost, ascompared to the ux evaluations. Our work-estimates can therefore be consideredto be good estimates of the actual computational work (and hence time) needed ona digital computer. We will assume that the di�erent ux-functions used (Roe andmodi�ed-Roe) required the same amount of work. This is a reasonable assumption.An e�cient implementation with the modi�ed Roe scheme and matrix precondition-ing combined should not require more than 10-15% more computational e�ort thanlocal time-stepping.We will initially concern ourselves with the details of the implementation of thenonlinear preconditioned residual P(U)Res(U) for the 2-D Euler equations.6.1 Ways of Implementing Local Preconditioning withMulti-Stage Time-SteppingIn general, a multi-stage update can be written as follows:U(0) = Un; (6.1)U(k) = U(0) +�t(k)R �U(k�1)� ; k = 1; ::;m; (6.2)Un+1 = U(m) (6.3)where R(U(k�1)) represents the preconditioned residual. There are two obvious waysto compute the preconditioned residual when implementing a multi-stage update witha nonlinear spatial operator.

1411. P is computed based on the most recent cell-centered state, i.e.R �U(k�1)� = P �U(k�1)�Res �U(k�1)� :This is the approach followed by Lee [33]. The multigrid convergence ratesobtained with this approach were the best among those obtained with the twoapproaches. Unless speci�ed otherwise, this is the approach we have used inour numerical studies. This method, though, is not as robust as approachnumber 2.2. P is computed based on the cell-centered state at the start of the update, i.e.R �U(k�1)� = P (Un)Res �U(k�1)� :This is the implementation of choice, particularly for complex ows incorpo-rating sonic and stagnation points. This approach is the most robust of thetwo described and is only marginally more expensive than the second approach(5% or so) with regards to the work required to obtain a converged solution.The di�erence in computational work required would reduce even further if wewere to take into account the fact that P does not need to be recomputed atevery stage of the update with this approach.6.2 Smoothing about a Sonic PointThe scaling parameter � = qj1 �M2j is present in the denominator of someterms in the preconditioner of Van Leer et al. as well as in the denominator ofcertain terms in the modi�ed Roe ux function. These terms become unbounded asM ! 1, at which point � ! 0.One obvious �x is to bound � away from zero by some arti�cial means, such asa smoothing function. Our implementation was as follows:

1420:00 0:70 1:40 2:100:000:701:402:10

M� M < 1M > 1curve �tFigure 6.1: Sonic-point smoothing function for �.We opted to �t a parabolic curve to � such that the �t matched the curvep1�M2 for a given value of M in both value and slope. The point of intersec-tion with the other branch, pM2 � 1, was also required to match in slope. Sinceonly three conditions can be speci�ed for this curve-�t, we opted not to specify thepoint of intersection with the curve pM2 � 1.Consider the curve-�t to have the formf(M) = �(M � (1 + �))2 + �min:Applying the conditions prescribed above and then solving the resulting algebraicequations (numerically) results in the following values for a prescribed intersectionpoint of M1 = 0:9: �min = 0:338, � = 10:888, � = �0:0052 and supersonic intersec-tion point M2 = 1:103. Note that �, which represents the deviation of the minimumfrom the sonic point is almost zero. The value of the curve-�t at M = 1 is 0:33829.These were the values that we used in our multi-grid computations of transonic andsupersonic channel ows with local preconditioning. Curve �ts with smaller valuesof �min lacked robustness for the �ne grid cases considered - the initial transients

143were prone to becoming unstable. However, residual-smoothing appears to alleviatethis problem (cf. Section 6.7).6.3 Smoothing about a Stagnation PointThe implementation of preconditioning in the computation of ows that containa stagnation point has so far been problematic. Even when M is smoothed awayfrom 0, non-convergence and instability have been reported [65]. We have thereforeavoided any test problems in which the steady solutions contain a stagnation point.Meanwhile, the robustness problem may have been solved by Darmofal and Schmid[13], by also considering the degeneration of the eigenvectors of P for M ! 0.6.4 Test Problem: Random Perturbations on a SquareMeshThe �rst test-case considered is the propagation out of the grid of a randomdisturbance to the freestream ow. A random perturbation was added to all theelements of the state vector in each cell (nonlinear disturbance). The grid used wasa 32 � 32 square-mesh and the freestream was assumed to have a ow angle of 10�with respect to one of the grid directions. The ghost cells across the boundaries wereset to freestream conditions (non-periodic BC).Table 6.1 shows the work required to reduce the magnitude of the �rst-orderresidual-norm (�rst-order spatial discretization) by �ve orders of magnitude. Four-stage time-stepping schemes were used in the computations. The keywords used areas follows: SG-LTS is single-grid with local time-stepping, SG-MTS is single-gridwith matrix time-stepping, MG-LTS is regular multigrid (4 levels) with local time-stepping and MG-MTS is regular multigrid (4 levels) with matrix time-stepping.As expected, local time-stepping performs poorly at low Mach numbers where the

144characteristic condition-number is high. The characteristic condition-number withlocal time-stepping is minimal at M = 0:5 and, indeed, local time-stepping performsbest at this Mach number. With M ! 1, the performance of local time-steppingdeteriorates again. Finally, when the Mach number becomes large and the charac-teristic condition-number tends towards one, local time-stepping approaches matrixtime-stepping in performance. Characteristic time-stepping performs well over theentire range of M, with signi�cantly faster convergence for supersonic ow than forsubsonic ow; this latter e�ect is due to the sweeping nature of error removal byupwind schemes for supersonic ow. Multigrid with matrix time-stepping performsextremely well across the range of Mach numbers considered. The work required isalmost constant, increasing marginally for some transonic/supersonic cases. Thereis clearly no extra convergence bene�t for supersonic ow; this is an artifact ofthe multigrid method as implemented. Standard multigrid has been designed withelliptical equations in mind and does not utilize the convective error-removal asso-ciated with hyperbolic equations. Even when an upwind relaxation scheme is used,its error-sweeping e�ects will be annihilated by standard prolongation. Downwind-biased prolongation has been advocated for supersonic ow by Koren and Hemker[29] and Leclercq and Stou�et [30].With MG-MTS, for most Mach numbers only 12 multigrid cycles were requiredto reduce the residual-norm by the required amount (1 cycle = 6.92 work units forthe 4-stage scheme). The speed-ups obtained on both single and multiple grids withrespect to local time-stepping justify the use of matrix time-stepping.Convergence studies with a second-order discretization have been performed forthe more complex test problem in the next section. In general, as the higher orderdiscretization is closer to the PDE, convergence speed-ups by preconditioning should

145Mach Number0.1 0.35 0.5 0.85 0.99 1.01 1.2 2.0 5.0SG-LTS 1224 408 352 984 1512 1552 784 272 164SG-MTS 280 272 272 272 224 188 160 148 128MG-LTS 996 401 263 263 816 1757 290 252 228MG-MTS 76 76 83 83 83 83 83 90 103Table 6.1: Work required to reduce norm of �rst-order residual by �ve orders ofmagnitude (to nearest work unit). � = 10�; �rst-order upwind operatorwith four-stage time-stepping. Description in the text.get closer to what is theoretically attainable for single-grid relaxation; this, in fact,has been observed [17, 16].Unlike in the next section, we did not attempt to �nd the \best" multigrid con-vergence rate by varying the number of stages and grid levels. Better multigridconvergence rates than those in Table 6.1 may therefore be attainable.Adding in explicit residual-smoothingThe test problem considered above was also solved with explicit residual-smoothingand the four-stage coe�cients presented in Section 5.3. In our implementation, theresidual smoothing is done at each stage of the multi-stage update. A reduction inthe work requirements was observed for all the cases considered. Table 6.2 shows thework requirements with explicit residual-smoothing. The single-grid improvements,in the order of a factor of 1:5� 2:0, are primarily due to the larger Courant numberallowed with residual smoothing. Multigrid improvements are observed as well butare not as large; in the best of the MG-MTS cases the work drops from 76 to 62

146Mach Number0.1 0.35 0.5 0.85 0.99 1.01 1.2 2.0 5.0SG-LTS 688 204 196 556 848 916 436 152 108SG-MTS 176 172 168 168 136 116 96 88 72MG-LTS 760 256 208 166 650 1438 180 180 187MG-MTS 62 62 62 69 69 69 69 76 83Table 6.2: Work required to reduce norm of �rst-order residual by �ve orders ofmagnitude (to nearest work unit). Discretization with explicit residualsmoothing and corresponding optimized four-stage time-stepping coe�-cients.work units (9 MG cycles), a decrease of 19%. This improvement in the convergencerate was achieved with a negligible increase in the computational expense.6.4.1 Grid-convergence studiesAll the results presented this far in this section have been based on the same 32�32 square mesh. We will now consider the e�ect of grid re�nement on the convergenceof MG-MTS. Two levels of re�nement are considered. We also consider two typesof boundary conditions: the in ow-out ow boundary conditions used earlier andperiodic boundary conditions in both coordinate directions. We have made use ofthree freestream Mach numbers for our tests: M = 0:35, 0:85 and 2:0.Periodic boundary conditionsUsing periodic boundary conditions negates the e�ects of convection and thereforeallows the study of the damping properties of the time-stepping scheme withouthaving to consider the e�ects of convection.

1470: 80: 160: 240:�6:0�4:0�2:00:0

work unitslog jresj Residual history 32x3264x64128x128Figure 6.2: Residual-history plots. M = 0:35, � = 10�. First-order spatial operator.MG-MTS with four-stage time-stepping scheme and 4�4 grid as coarsestgrid level. Periodic boundary conditions.Figures 6.2 to 6.4 show the normalized residual-history plots for the three freestreamMach number cases considered.In the subsonic case (Figure 6.2) there is only a factor 1.20 variation in thework required to reduce the residual norm by 5 orders of magnitude. However, theasymptotic convergence rates vary much more: from 70 work units (32 � 32 grid)to 110 work units (128 � 128 grid) per order of magnitude residual reduction, or afactor 1.6.For the transonic case (M = 0:85, Figure 6.3), the spread in the work increasesto a factor of 1.27, and the asymptotic convergence rates now vary by a factor 1.8.The performance variation is much more dramatic for the supersonic case (M =2:0, Figure 6.4). The work very nearly doubles from coarsest to �nest grid, while theasymptotic convergence rate decreases by a factor 2.6.It is clear that grid-independent convergence has not been realized with the cur-

1480: 80: 160: 240:�6:0�4:0�2:00:0

Work unitslog jresj Residual history 32x3264x64128x128Figure 6.3: Residual-history plots. As Figure 6.2 but for M = 0:85.

0: 200: 400: 600:�6:0�4:0�2:00:0work unitslog jresj Residual history 32x3264x64128x128

Figure 6.4: Residual-history plots. As Figure 6.2 but for M = 2:0.

1490: 40: 80: 120:�6:0�4:0�2:00:0

work unitslog jresj Residual history 32x3264x64128x128Figure 6.5: Residual-history plots. M = 0:35, � = 10�. First order spatial operator.MG-MTS with four-stage time-stepping scheme and 4�4 grid as coarsestgrid level. In ow-out ow boundary conditions.rent multigrid strategy. Note that the total work required, while hardly dependenton M for the 32 � 32 grid, approximately doubles when going from M = 0:35 toM = 2:0 on the 128 � 128 grid.In ow-out ow boundary conditionsFigures 6.5 to 6.7 show the normalized residual-history plots for the three freestreamMach number cases considered with in ow-out ow boundary conditions, which al-lows for convective error removal.One expected e�ect of these boundary conditions is to make the multigrid con-vergence rate more dependent on the grid size, since the multigrid strategy partlynegates the convective error removal. On the other hand, the convective error re-moval itself should at least reduce the work required on some grids, for some Machnumbers considered.

1500: 40: 80: 120:�6:0�4:0�2:00:0

work unitslog jresj Residual history 32x3264x64128x128Figure 6.6: Residual-history plots. As Figure 6.5 but for M = 0:85.

0: 80: 160: 240:�6:0�4:0�2:00:0work unitslog jresj Residual history 32x3264x64128x128

Figure 6.7: Residual-history plots. As Figure 6.5 but for M = 2:0.

151From Figures 6.5 - 6.7 it is seen that the total work required on the three gridsindeed shows somewhat more spread for all the three Mach numbers considered, i.e.for M = 2:0, (Figure 6.7), the spread is a factor 2.2. The asymptotic convergencerates show the e�ect much more strongly: the variation is now a factor 4.4.Is is also seen, though, that convergence is faster on all grids for all Mach numbers;the speed-ups are of the order of a factor 2 compared to the case with periodicboundary conditions.We may conclude that our approach works reasonably well for the subsonic case,being fairly close to grid-independence. (An asymptotic rate reduction at M = 0:35of a factor 1.6 for an increase in number of unknowns N of a factor 16 means thatthe work varies as N0:17). However, the convergence rates become more stronglydependent on the size of the grid as the freestream Mach number increases, (work� N0:53 asymptotically for M = 2:0). This clearly indicates that modi�cations tothe multigrid method are due: in particular, the hyperbolic nature of the equationsmust be exploited.This does not negate the fact that MG-MTS is the fastest of the approaches thatwe have considered, i.e. MG-MTS performs best on all the grids considered.6.4.2 Importance of the optimal multi-stage coe�cientsWe now consider the importance of the optimal multi-stage coe�cients obtainedin Chapter IV: are they required in order to achieve the convergence rates obtainedearlier with MG-MTS?For our tests, we chose the subsonic case used earlier (M = 0:35, � = 0�, 32� 32mesh, four-stage scheme with four grid-levels, �rst-order operator and in ow-out owboundary conditions). We �rst replaced the optimal-coe�cients used earlier (from

152Table 4.1) with Tai's coe�cients, which are sub-optimal. The work required increasedfrom 76 work-units to 97 work-units, an increase of 29%. We then chose coe�cientshalfway between Tai's and the optimal coe�cients in (~�; � space). The solution thenrequired 90 work units, an increase of 18%. Using coe�cients even closer to theoptimal scheme, obtained by halving the distance in parameter space again, required83 work units, an increase of 9%. It is clear that, for this perturbation direction,using the optimal coe�cients produces the best convergence rate. We did not �nd itmeaningful to study the e�ect of perturbations in random directions on convergence.It is our experience that the damping properties of the multi-stage scheme are verysensitive to perturbations in the Courant number, more so than the fractional time-steps �i. We therefore studied the e�ect of increasing and decreasing the optimalCourant number by 15%. Such variations in the Courant number are easily obtainedif the length-scale used is not chosen appropriately. For the case where the Courantnumber was increased by 15%, the work required increased to 166 work units, anincrease of 118%. The speed-up in convective error-removal is countered by thedecrease in damping for this case. For the case where the Courant number wasdecreased by 15%, the work required increased to 90 work units, an increase of 18%.Our experiments therefore support the conclusion that the optimal coe�cientsderived earlier are indeed necessary in order to obtain the best convergence rateswith this approach.6.4.3 Comparison of damping rates per multigrid cycleFinally, we compared the asymptotic convergence rates for two multigrid cy-cles, one incorporating local time-stepping and the other incorporating matrix time-stepping. We chose the subsonic case considered above (M = 0:35) on the 32 � 32

153square mesh, with periodic boundary conditions to avoid convective error removal.We used the optimal four-stage coe�cients from Table 4.1 for both implementations.Both multigrid cycles have four grid levels.Ideally, if we were to ignore the e�ect of the restriction and prolongation oper-ators, including aliasing, and if we solved the problem exactly on the coarsest grid,we would expect the convergence rate to be bounded from below by the value of �for this set of coe�cients, which equals 0.2349. This is equivalent to one order ofmagnitude residual-norm reduction per 1.6 multigrid cycles.The asymptotic convergence rate with local time-stepping turned out to be veryclose to unity, around 0.992. We obtained a value of 0.796 with local preconditioning,which corresponds to one order of magnitude residual-norm reduction in 10 cycles,i.e. only one-sixth of the theoretically attainable limit.6.5 Test Problem: Flow past a Bump in a ChannelWe will now consider the solution to the problem of inviscid ow through achannel with a non-smooth bump. This ow shows strong grid-alignment and istherefore considered to be a good test-case for semi-coarsened multigrid. Indeed,regular multigrid relaxation failed to converge when applied to a second-order spatialdiscretization. With the more di�usive �rst-order discretization there appeared tobe enough coupling between streamlines to overcome the alignment problem.The bump is a circular arc with a thickness of 4:2% of the chord. The lengthof the channel is 5.5 chords, with height 2 chords. Solutions were obtained withtwo di�erent grids, generated algebraically by slightly deforming uniform squaremeshes. A 128 � 64 grid was used to obtain most of the solutions presented in thenext few subsections. In order to save computing time, the comparison of multigrid

154and single-grid methods was carried out on a 64 � 32 grid. Freestream conditionswere utilized as initial values and for the virtual cells beyond the in ow/out owboundaries. Re ecting wall-boundary conditions were applied at the upper and lowerwalls.In the following three sub-sections, we compare the best possible single and multi-grid convergence rates. We have solved the test-problem with di�erent grid levelsand varying number of stages in the time-stepping scheme. We made use of Tai'smulti-stage coe�cients [56] for the cases with local time-stepping. Tai's coe�cientsappear to be more robust for multigrid with local time-stepping, for this test-caseat least. We used the multi-stage coe�cients of Section 4.5 for the cases of localpreconditioning with and without regular multigrid. The coe�cients obtained inSection 4.6 were used for cases solved with local preconditioning and semi-coarsenedmultigrid. The varying length-scale described in Section 4.4 was used in comput-ing the time-step. It should be noted that the convergence-rates obtained with localtime-stepping are fairly insensitive to this length-scale de�nition. On the other hand,the convergence-rates with local preconditioning are quite sensitive to the varyinglength-scale. Using a length-scale independent of ow-angle can decrease the ob-served convergence rates by 30% or more for these cases. The convergence resultsare summarized in Section 6.5.4.Second-order solutions were obtained using a spatial discretization with � = 0together with Van Albada's ux limiter. Defect-correction multigrid cycles wereused for the multigrid cases (cf. Section 2.4). Nested iteration was used to improvethe initial solution; the work required for nested iteration is counted in the resultspresented.We used a norm based on an estimate of the truncation error taken from Mulder

155[42], as our convergence check for second-order solutions. This decision was basedon the observation that this norm was a better indicator of convergence rate thanthe norm of the second-order residual, which has been known to hang even while thesolution is still converging. The norm was de�ned as follows:kTEk1 = kRh1(Uh)k1kI2hh Rh1(Uh)�R2h1 (I2hh Uh)k1 ; (6:4)where the subscript 1 on the residual terms indicates that we are considering the �rstcomponent of the residual-vector (continuity). It should be noted that use of thisnorm makes direct comparison of the work required to obtain �rst and second-ordersolutions di�cult. In general, for the values of kTEk1 that we have used, the residualnorm drops by about three and a half orders of magnitude. It should be pointed outthat the solutions presented also used the same value of the TE norm to signify thatthe solution had converged. The reader is referred to [42] for a further discussion onusing estimates of this type.6.5.1 Subsonic caseWe chose a Mach number ofM = 0:35 to be representative of the subsonic regime.The standard Roe scheme with local time-stepping performs poorly with regard toconvergence at low Mach numbers (less than 0.1 or so), so it is fair to say that thespeed-ups observed with this moderate value of M are conservative estimates of thespeed-ups that local preconditioning can achieve together with multigrid at low Machnumbers. The solutions that may be obtained with characteristic time-stepping andmodi�ed Roe ux at very low Mach numbers are also more accurate than thoseobtained with the standard Roe ux for the otherwise same spatial discretization(cf. [33, 63, 16, 17]). Figure 6.9 shows a second-order solution obtained with localpreconditioning on the 128 � 64 grid. As can be noted from the �gure, the solution

1560: 500: 1000: 1500:�8:0�4:00:04:0 work unitslog jjresjj 1st order spatial operator, 64x32 grid single+presc mg+locsc mg+prereg mg+locreg mg+pre

Figure 6.8: Residual-history plots for the transonic channel case (M = 0:85). 64x32grid. First-order spatial di�erencing. Four-stage time-stepping schemes;8x4 grid taken as coarsest grid for multigrid. The convergence historyfor a single-grid with local time-stepping is not shown; it lies almost ontop of the curve for the single-grid preconditioned scheme.is fairly symmetric about the bump. The entropy layer that is generated in the wakeis observed with all the ux functions that are based on one-dimensional physics (cf.[38] for some recent developments in multi-dimensional upwinding).Table 6.4 shows the work units required to obtain a practically converged �rst-order solution to the problem with the semi-coarsened multigrid method and a vary-ing number of grid levels. For comparison, the work units required with regularmultigrid for the same case are presented in Table 6.3. As can be observed fromthe latter table, the best combination of number of grid-levels and number of stagescan be hard to predict, especially with local time-stepping and regular multigrid.One reason could be that the multi-stage schemes used with local time-stepping areunable to e�ciently damp all the high-frequency error-modes that are present inthe transient solution. In contrast, when used with semi-coarsening, the multi-stage

157scheme is only required to be a good damper of high-high frequency errors. Tables6.4 and 6.3 have di�erent single-grid convergence results for matrix preconditioningdue to the fact that two di�erent sets of multi-stage coe�cients were used here; oneoptimized for use with regular multigrid and the other with semi-coarsenedmultigrid.With matrix preconditioning, more often than with local time-stepping, a pointeventually is reached where increasing the number of grid levels or number of stagesbecomes more expensive - the extra work outweighs the improvement in the conver-gence rate, if any. This \saturation e�ect" is noticable in the other tables presentedas well. This may have to do with the large stencil of the multi-stage schemes arecompared to the size of the coarser grids. As a rule, a 4 or 5 stage scheme and 4 gridlevels would be expected to provide good convergence rates for computations on a64 � 32 grid.On a single grid, matrix preconditioning o�ers the double bene�t of improvingsingle-grid convergence as well as providing robustness through the use of schemeswith improved high-frequency damping of error-modes. This improved high-frequencydamping also translates into improved multigrid performance. Tables 6.3 - 6.5 showboth e�ects. The single-grid speed-up from local to matrix time-stepping is mod-est, but in Table 6.3, for instance, multigrid relaxation with matrix time-steppingreduces the work by a factor 13, as compared to a factor 3.3 for multigrid with localtime-stepping.6.5.2 Transonic caseWe took M = 0:85 to be the freestream Mach number for the transonic case.Figure 6.10 shows a second-order solution obtained with local preconditioning onthe 128 � 64 grid. Figure 6.8 shows typical residual-history plots for �rst-order

158�1:5 0:5 2:50:02:0 0

t=c = 0:042, modi�ed Roe scheme with � = 0. 2 0.3074 0.3176 0.3278 0.33710 0.34712 0.35714 0.36716 0.37718 0.387mn 0.302mx 0.388Figure 6.9: Mach number contours. Flow past a non-smooth bump in a channel.M = 0:35, t=c = 0:042, 128 � 64 grid, spatial discretization with � = 0,Van Albada's limiter and modi�ed Roe ux.Number of stagesLocal time-stepping Matrix preconditioning2 3 4 5 6 2 3 4 5 6No. l 3022 3015 2940 2940 2940 2204 2178 2136 2130 2160of 2 1346 1120 1001 936 892 697 586 557 482 423grid 3 1535 1307 1181 1109 1064 563 330 206 166 165levels 4 1579 1351 1224 1152 1107 210 179 166 165 1715 1595 1368 1242 1163 1118 255 186 168 166 173Table 6.3: Work units required to reduce the norm of the �rst-order residual by �veorders of magnitude. M = 0:35 test-case. Regular multigrid. 64 � 32grid.

159Number of stagesLocal time-stepping Matrix preconditioning2 3 4 5 6 2 3 4 5 6No. l 3022 3015 2940 2940 2940 2706 2661 2568 2520 2670of 2 2087 1700 1525 1440 1349 1784 1663 1601 1368 1536grid 3 1779 1451 1324 1232 1190 1369 961 702 558 502levels 4 1795 1459 1303 1207 1123 404 316 279 268 2735 2035 1662 1454 1355 1271 421 320 301 288 293Table 6.4: Work units required to reduce the norm of the �rst-order residual by�ve orders of magnitude. M = 0:35 test-case. Semi-coarsened multigrid.64� 32 grid. Number of stagesLocal time-stepping Matrix preconditioning3 4 5 6 3 4 5 6Number l 3702 3740 3685 3744 3126 2976 2380 2916of 2 2241 1622 1284 1087 1366 1271 780 838grid 3 692 582 582 583 388 301 632 615levels 4 755 751 705 708 319 191 194 219Table 6.5: Work units required to reduce kTEk1 to 10�2. M = 0:35 test-case.Second-order spatial operator (� = 0). Defect-correction cycles used to-gether with nested iteration for initial guess. Semi-coarsened multigrid.64� 32 grid.

160spatial di�erencing on the 64 � 32 grid; all runs in this �gure were made with four-stage schemes, and multigrid runs were made with four grid-levels (8x4 as coarsestgrid). The slow, oscillatory decay of the residual norm experienced with the single-grid preconditioned scheme (local time-stepping, which is not shown for the singlegrid, gives an almost indistinguishable residual history) reveals the presence of asmaller time-scale, viz. the cross-channel travel time for acoustic waves. Note thatthe convergence histories for semi-coarsened multigrid are smoother than those forregular multigrid, indicating greater robustness. Tables 6.6, 6.7 and 6.8 give the workunits required to obtain a converged solution with regular multigrid, semi-coarsenedmultigrid (�rst order) and semi-coarsened multigrid (second order). In this case, noimprovement by preconditioning is observed on a single grid; on the contrary, allcalculations except one take longer than with local time-stepping. This appears tobe a problem inherent to this test case with parallel re ecting walls: we shall explorethis further in Sections 6.5.4 and 6.6).Semi-coarsened multigrid performed poorly with local preconditioning for thesecond-order case when computed using one or two grid-levels, with convergencegetting hung up in a limit cycle. We suspect this to be an e�ect of the particularsonic-point smoothing that was used. The smoother modi�es the damping and wave-propagation properties of the time-stepping scheme around a sonic point in a waythat is hard to analyze nonlinearly. The e�ect was not observed when more grid levelswere used in the computation; apparently the corrections from the coarser grids, withless resolution, remedied the problem. The 4-grid results are again excellent.Tables 6.6 - 6.8 again show the bene�cial e�ect of preconditioning on multigridrelaxation; in Table 6.6, for instance, the speed-up on multiple grids with matrix-timestepping is a factor 23, as compared to 5.5 with local time-stepping.

161�1:0 1:0 3:00:02:0

t=c = 0:042, modi�ed Roe scheme with � = 0. 2 0.6804 0.7606 0.8408 0.92010 1.00012 1.08014 1.16016 1.240mn 0.618mx 1.293Figure 6.10: Flow past a non-smooth bump in a channel. M = 0:85, t=c = 0:042,128�64 grid. Spatial di�erencing with � = 0, Van Albada's limiter andmodi�ed Roe ux. Number of stagesLocal time-stepping Matrix preconditioning2 3 4 5 6 2 3 4 5 6No. l 4140 4140 4140 4140 4140 4622 4497 4276 4150 4068of 2 1346 1144 1041 983 937 2319 1627 1155 922 769grid 3 1010 878 814 776 752 1289 374 200 182 174levels 4 1069 919 865 823 801 245 207 191 189 1815 1621 1441 1361 1313 1253 337 283 230 216 193Table 6.6: Work units required to reduce the norm of the �rst-order residual by �veorders of magnitude. M = 0:85 test-case. Regular multigrid. 64 � 32grid.

162Number of stagesLocal time-stepping Matrix preconditioning2 3 4 5 6 2 3 4 5 6No. l 4140 4140 4140 4140 4140 5194 5019 4924 4920 5118of 2 2613 2250 2074 1962 1888 4115 3163 2760 2358 2324grid 3 1766 1468 1344 1256 1216 1779 1013 662 535 449levels 4 1294 1065 977 912 911 453 355 326 349 3645 1614 1321 1203 1124 1075 1509 746 401 404 424Table 6.7: Work units required to reduce the norm of the �rst-order residual by�ve orders of magnitude. M = 0:85 test-case. Semi-coarsened multigrid.64� 32 grid. Number of stagesLocal time-stepping Matrix preconditioning3 4 5 6 3 4 5 6Number l 5571 5400 5305 5508 18177 11688 9600 12006of 2 5933 4205 3277 3132 x x x xgrid 3 1912 1524 1220 1052 2605 937 338 538levels 4 711 515 540 702 260 191 167 189Table 6.8: Work units required to reduce to reduce kTEk1 to 5 � 10�2. � = 0spatial discretization. Defect-correction cycles used together with nestediteration for initial guess. M = 0:85 test-case. Semi-coarsened multigrid.64� 32 grid.

163�1:0 1:0 3:00:02:0

t=c = 0:042, modi�ed Roe scheme with � = 0. 2 1.0814 1.1646 1.2478 1.33010 1.41312 1.49514 1.578mn 1.039mx 1.660Figure 6.11: Flow past a non-smooth bump in a channel. M = 1:4, t=c = 0:042,128 � 64 grid. Spatial discretization with � = 0, Van Albada's limiterand modi�ed Roe ux.6.5.3 Supersonic caseWe took M = 1:4 to be the freestream Mach number for this case. Figure 6.10shows a second-order solution obtained with local preconditioning on the 128 � 64grid. For comparison we also show the solution obtained on the 64�32 grid used forthe convergence experiments (Figure 6.12). Tables 6.6, 6.7 and 6.8 give the numberof work units required to obtain a converged solution with regular multigrid, semi-coarsened multigrid (�rst order) and semi-coarsened multigrid (second order). In allthese cases the single-grid speed up was evident, in accordance with the convectivecharacter of supersonic steady ow. Standard multigrid could not really improvemuch upon this.In the next section we try to draw some general conclusions from the collectionof results in Sections 6.5.1 to 6.5.3.

164�1:0 1:0 3:00:02:0

t=c = 0:042, modi�ed Roe scheme with � = 0. 2 1.1704 1.2506 1.3308 1.41010 1.49012 1.570mn 1.094mx 1.619Figure 6.12: Flow past a non-smooth bump in a channel. M = 1:4, t=c = 0:042,64 � 32 grid. Spatial discretization with � = 0, Van Albada's limiterand modi�ed Roe ux. Number of stagesLocal time-stepping Matrix preconditioning2 3 4 5 6 2 3 4 5 6No. l 1550 1551 1552 1550 1554 296 294 288 300 306of 2 752 638 580 548 530 232 192 176 161 161grid 3 734 634 580 547 514 204 168 167 174 174levels 4 760 662 609 576 544 210 179 180 181 1815 765 667 614 582 549 225 187 181 183 183Table 6.9: Work units required to reduce the norm of the �rst-order residual by �veorders of magnitude. M = 1:4 test-case. Regular multigrid. 64� 32 grid.

165Number of stagesLocal time-stepping Matrix preconditioning2 3 4 5 6 2 3 4 5 6No. l 1550 1551 1552 1550 1554 380 375 364 355 378of 2 956 825 763 720 706 283 250 244 234 249grid 3 890 743 682 651 661 329 304 281 302 318levels 4 954 828 791 778 759 404 355 326 322 364Table 6.10: Work units required to reduce the norm of the �rst-order residual by�ve orders of magnitude. M = 1:4 test-case. Semi-coarsened multigrid.64 � 32 grid. Number of stagesLocal time-stepping Matrix preconditioning3 4 5 6 3 4 5 6Number l 1422 1384 1365 1344 420 420 425 414of 2 895 933 781 753 356 350 332 309grid 3 724 898 722 769 361 370 360 358levels 4 809 1047 993 732 436 399 380 400Table 6.11: Work units required to reduce to reduce kTEk1 to 10�2. � = 0 spatialdiscretization. Defect-correction cycles used together with nested iter-ation for initial guess. M = 1:4 test-case. Semi-coarsened multigrid.64 � 32 grid.

1666.5.4 Summary of results - channel problemThe convergence results presented for the channel problem are summarized belowin Tables 6.12, 6.13 and 6.14. These summarized results represent the solutionsobtained with minimum work for each case.Tables 6.13 and 6.14 indicate that local and matrix time-stepping require similaramounts of work in the single-grid subsonic and transonic runs. This is probablya peculiarity of the problem. (This behavior is not observed when solving for the ow in a semi-in�nite channel - see Section 6.6). The re ective wall-boundary con-ditions in the channel provide minimal attenuation of vertically moving acousticerror-modes; this e�ect dominates single-grid convergence. This is not a problemfor the supersonic cases, which have a largely convective nature. Regardless of thesingle-grid performance, the speed-up of multigrid with local preconditioning overmultigrid with local time-stepping is impressive; the former is 3-4 times faster in allcases. Multigrid convergence with local preconditioning in the subsonic and tran-sonic cases is also an order of magnitude faster than in the corresponding single-gridcases. Local preconditioning performs admirably on a single-grid for the supersoniccase, and there is not much improvement possible without modifying multigrid tobetter handle convection-dominated ow.A semi-coarsened multigrid cycle is more expensive than a regular multigrid cycle(a factor of three on average for the cases considered - this factor depends on thenumber of grid levels chosen). Each semi-coarsened multigrid cycle is able to reducethe residual norm by a larger factor than a corresponding regular multigrid cycle.However some of the extra overhead that is inherent in the method does show up inthe work required to obtain the same solution with the same number of grid levels andstages in the time-stepping scheme. Tables 6.12 and 6.13 re ect the di�erence in work

167M = 0:35 M = 0:85 M = 1:4Local TS Matrix TS Local TS Matrix TS Local TS Matrix TSSingle 2940 2130 4140 4150 1550 288Multigrid 892 165 752 174 514 161Table 6.12: Comparison of �rst-order convergence rates for ow past a semi-circularbump in a channel, 64x32 grid. Work units required to reduce the normof the residual by �ve orders of magnitude. (Local TS � local time-stepping, Matrix TS � local preconditioning). Regular multigrid.M = 0:35 M = 0:85 M = 1:4Local TS Matrix TS Local TS Matrix TS Local TS Matrix TSSG 2940 2520 4140 4920 1550 355SC-MG 1123 268 911 326 651 234Table 6.13: Comparison of �rst-order convergence rates for ow past a semi-circularbump in a channel, 64x32 grid. Work units required to reduce the normof the residual by �ve orders of magnitude. Semi-coarsened multigrid.M = 0:35 M = 0:85 M = 1:4Local TS Matrix TS Local TS Matrix TS Local TS Matrix TSSG 3685 2380 5305 9600 1344 414SC-MG 582 191 515 167 722 309Table 6.14: Comparison of second-order (� = 0) convergence rates for ow past asemi-circular bump in a channel, 64x32 grid. Work units required toreduce kTEk1 to 10�2 for M = 0:35 and M = 1:4 and to 5 � 10�2 forM = 0:85. Nested iteration with 5 defect-correction sweeps on eachcoarse grid level was used initially to improve robustness for multigridsolutions. Semi-coarsened multigrid.

168�1:0 1:0 3:00:02:0

t=c = 0:042, modi�ed Roe scheme with � = 0. 2 0.7204 0.8006 0.8808 0.96010 1.04012 1.12014 1.200mn 0.642mx 1.223Figure 6.13: Transonic ow past a non-smooth bump on a wall in a semi-in�nitedomain. M = 0:85, t=c = 0:042, 128 � 64 grid. Spatial discretizationwith � = 0, Van Albada's limiter and modi�ed Roe scheme.required. It should be pointed out though that the semi-coarsened multigrid methodmakes up for the extra work required in the form of increased robustness. Resid-ual history plots indicate that the convergence rate with regular multigrid may varyerratically, especially in the transonic regime, whereas the convergence history forsemi-coarsening is smooth. Regular multigrid also performed poorly when attempt-ing to obtain second-order solutions, particularly for the transonic case considered.Several of the combinations of number of stages and grid levels resulted in divergentsolutions or limit cycles, for both local time-stepping and matrix time-stepping. Wetherefore did not attempt to create a table with second-order solutions incorporatingregular multigrid for this test case.6.6 Test Problem: Transonic Flow past a Bump on a Wallin a Semi-in�nite DomainAs noted earlier, poor single-grid convergence rates were obtained for the subsonicand transonic channel cases considered in the previous section. There was little orno speed-up observed when using matrix time-stepping on a single grid. This was

169First Order Second OrderLocal TS Matrix TS Local TS Matrix TSSingle 2912 1204 1384 692SC-MG 907 349 827 249Table 6.15: Comparison of �rst-order and second-order convergence rates for owpast a semi-circular bump on a wall, 64x32 grid, M = 0:85. Work unitsrequired to reduce the norm of the residual by �ve orders of magnitude(for �rst order) and kTEk1 to 5� 10�2 (second order). Four-stage time-stepping schemes were used for all runs. Four grid-levels were used forthe multigrid runs. (Local TS � local time-stepping, Matrix TS � localpreconditioning). Semi-coarsened multigrid.thought to be due to the wall-boundary conditions providing minimal attenuation ofacoustic error-modes.To test this hypothesis, we chose to solve for the ow past a bump on a wall in asemi-in�nite domain. The length of the computational domain was taken to be 5.5chords and the height 3. Free-stream boundary conditions were applied at all non-wall boundary conditions. As before, the bump was a circular arc with a thicknessof 4:2% of the chord. A 128 � 64 grid was used to obtain the solution presented inFigure 6.13. A 64� 32 grid was used to study the convergence rates attainable withmultigrid and local preconditioning.We did not undertake an exhaustive study in this case. Rather, we chose M =0:85, the worst case, and solved the problem with �rst- and second-order accuracy.Figure 6.13 shows a second-order solution that was obtained on the 128 � 64 gridfor this case. All our tests were done with four-stage time-stepping schemes. Themultigrid runs were done with four grid-levels (8 � 4 coarsest grid level). Table6.15 shows that matrix time-stepping is at least twice as fast as local time-stepping

170�1:0 1:0 3:00:02:0

t=c = 0:042, modi�ed Roe scheme with � = 0 and explict RS. 2 0.7004 0.8006 0.9008 1.00010 1.10012 1.20014 1.30016 1.400mn 0.638mx 1.404Figure 6.14: Flow past a non-smooth bump in a channel. M = 0:99, t=c = 0:042,128 � 64 grid. Spatial discretization with � = 0, Van Albada's limiterand modi�ed Roe ux. Explicit residual-smoothing was used to obtainthis solution.on the single grid runs for this problem. It would therefore appear that the poorsingle-grid convergence rates obtained in the previous section are indeed due to theadditional re ecting wall. It should be noted that the sonic-point smoother probablystill reduces the convergence rates for the runs with local preconditioning. However,it is required for robustness.The results in Table 6.15 should not be compared to those in Tables 6.13 and6.14 as they have been obtained under di�erent conditions; no nested iteration wasused here in obtaining the second-order solutions and all the solutions in this casewere obtained with four-stage time-stepping schemes (and four grid-levels for themultigrid cases), unlike before.6.7 Improving Robustness with Explicit Residual Smooth-ingThe preconditioner of Van Leer et al. is known to be lacking in robustness aboutsonic and stagnation points. The stagnation-point singularity appears to be quite a

171di�cult problem to correct and there has been substantial research into achievingrobustness under these conditions (cf. [65, 13]).The sonic-point singularity may be overcome more easily. Its cause is the scalingparameter � = qj1 �M2j, present in the denominator of some terms in the pre-conditioner of Van Leer et al. as well as in the denominator of certain terms in themodi�ed Roe ux function. These become unbounded as M ! 1, at which point� ! 0.One obvious �x, utilized in the calculations of Sections 6.5-6.6 is to bound � awayfrom zero by some arti�cial means, such as a smoothing function (cf. Section 6.2).With this approach and with the additional application of explicit residual smoothing(cf. Section 5.3), we were actually able to obtain a solution to the channel problemwith freestream Mach numbers very close to 1. Figure 6.7 shows a solution obtainedfor M = 0:99; we also obtained a solution for M = 1:01 with ease, and even fora sonic freestream, i.e. M = 1, (although the physical existence of such a steadysolution may be doubtful). The three solutions look almost identical.

CHAPTER VIICONCLUSIONSIn this thesis, we have shown that combining local preconditioning with multi-grid relaxation makes the multigrid method very e�cient in obtaining steady-statesolutions to the two-dimensional Euler equations of uid ow.Key to the success of this combination is the development of single-grid marchingschemes with guaranteed high-frequency damping. An optimization formulation hasbeen described that may be used to obtain multi-stage schemes with superior damp-ing; the optimization is taken over the high-frequency content in the Fourier footprintof the preconditioned spatial operator. Both standard multigrid and semi-coarsenedmultigrid have been considered, requiring optimization over di�erent frequency do-mains. The optimization problem has been solved by the method of simulated an-nealing together with the downhill-simplex method. This solution algorithm didnot require frequent restarts, unlike methods we used in earlier attempts. Tables ofmulti-stage coe�cients have been presented that are based on the solution to thisoptimization problem.We have demonstrated that the combination of local preconditioning and multi-stage time-stepping can produce relaxation schemes that boast strong high-frequencydamping for the entire range of ow angles, Mach numbers, cell aspect-ratios and (for172

173Navier-Stokes operators) cell-Reynolds numbers. Such schemes are ideally suited foruse as single-grid relaxation schemes in a multigrid relaxation framework, particularlyif semi-coarsening is used. In addition they are superior relaxation schemes if only asingle grid is used, in comparison to other explicit marching schemes with or withoutlocal preconditioning. The preconditioning itself already accelerates the convergenceto a steady solution, and the high-frequency damping provides robustness.Numerical studies on a rather coarse two-dimensional grid (64�32) indicate thatmultigrid speed-ups of a factor of 3-4 may be obtained when local preconditioning isused, as compared to local time-stepping. This amply o�sets the extra cost of includ-ing the preconditioning matrix in the solver, which is estimated at 10-15%. Studiesalso indicate that explicit residual-smoothing can further improve convergence ratesup to 25%, as well as improving robustness, with only a minimal increase in thecomputational e�ort required per update. Important in achieving good convergencerates is to model correctly the variation in the e�ective cell dimension (a length-scaleentering in the optimal Courant number) with ow angle. Equally important is tooptimize the multi-stage coe�cients for the full spatial Euler operator. Using othersets of optimal coe�cients, such as Tai's [56], optimized for 1-D convection operators,can worsen convergence rates by around 25%.The extension to Navier-Stokes operators and three space dimensions, and theimplemention on unstructured meshes have also been brie y considered.7.1 Future WorkIssues that remain to be resolved include the following.1. It is unclear what is a good choice for the value of the prescribed dampingwhen obtaining multistage coe�cients with a prescribed damping value, as in

174Section 5.2. This is primarily an issue when obtaining multi-stage coe�cientsfor Navier-Stokes operators. More multigrid experimenting is needed to deter-mine the best balance between parameters such as damping rate and time-stepvalue. The problem can be avoided by solving a more di�cult constrainedoptimization problem.2. The issue of robustness of preconditioned schemes at a stagnation point remainsto be resolved. Recent research [65, 13] indicates that a modi�cation of thepreconditioner may be needed at very low Mach numbers. It may also bepossible to derive a better �x for the sonic-point singularity than the sonic-point smoother that we have described in Section 6.2. This smoother appearsto a�ect convergence rates in some cases, particularly for higher-order solutions.3. It may be possible to improve the single-grid (and possibly multigrid) con-vergence rates for the transonic channel case (Section 6.5) by using soft wallboundary conditions such as those due to Roe and Mazaheri [50, 37]. Soft-wall boundary conditions and other techniques such as bulk-viscosity damp-ing [37], are convergence-accelerating mechanisms not related to characteristictime-stepping; it may therefore be possible to improve convergence rates bycombining these techniques.4. It appears that Navier-Stokes preconditioners require a modi�cation to the con-tinuity equation in order to be suited for use at very low Re�x. One suggestion[70] has been to add in an element of order � 1=Re�x on the main diagonal.5. It is unclear how the large cell-aspect ratios induced by semi-coarsening on amesh meant to resolve boundary layers would a�ect convergence.

1756. Also, semi-coarsened multigrid has not yet been fully developed for three-dimensional PDEs. If all semi-coarsened grids were to be included in the cycle,the total work would grow unboundedly as more and more grid levels were in-troduced (for problems with �ner and �ner base grids). Current research pointsin the direction of using \sparse grids", i.e. using a subset of semi-coarsenedgrids on which all error components can be represented [20].Other, broader topics of interest that require further work include:1. Multigrid Euler solutions with local preconditioning on an unstruc-tured mesh. It would be interesting to see how well the multi-stage coe�cientspresented in this thesis would work in an unstructured multigrid formulationthat incorporates local preconditioning. It is our hope that, with proper scal-ing of the Courant-number length-parameter, these coe�cients would providegood high-frequency damping and good convergence rates. It remains to beseen if the speed-ups over multigrid with local time-stepping observed here (3-4times) carry over to unstructured meshes.2. Multigrid Euler solutions with local preconditioning on a parallelprocessor. The method as described may be ported very easily to a parallelprocessor using a domain decomposition method. Some preliminary exper-iments that we undertook on the KSR-1 disributed-shared memory parallelprocessor showed an almost linear speedup with an increase in the numberof processors. Dynamic load-balancing of a multigrid algorithm such as semi-coarsening while maintaining locality of reference is a non-trivial problem.3. Modifying multigrid to better handle convection-dominated prob-lems. As mentioned earlier, standard multigrid has been designed with el-

176liptical equations in mind and does not utilize the convective error-removalthat is associated with hyperbolic equations. Even when an upwind relaxationscheme is used, its error-sweeping e�ects will be annihilated by standard pro-longation. Downwind biased prolongation has been advocated for supersonic ow by Koren and Hemker [29] and Leclercq and Stou�et [30].

APPENDICES

177

178APPENDIX AThe Method of Simulated AnnealingSimulated annealing is a powerful technique that has been used with success insolving optimization problems, particularly those where a desired global extremumis hidden among many local extrema.Central to the method of simulated annealing is an analogy with statistical me-chanics, speci�cally with the way that liquids freeze and crystallize or metals cooland anneal. As noted by Kirkpatrick et al. [27], physical systems may be coaxedinto a minimum energy con�guration, such as that of a crystal, by a slow annealingapproach. At high enough temperatures, the molecules of a liquid move freely withrespect to one another. If the liquid is cooled slowly, as thermal mobility is lost theatoms are often able to line up and form a pure crystal that is a minimum energystate for this system. If a liquid metal is cooled quickly, or \quenched", it does notreach this state, but rather ends up in a state with somewhat higher energy.The essence of the process of reaching a minimum energy state is therefore slowcooling, allowing su�cient time for the redistribution of atoms as they lose mobility.O�ered a choice of options, a simulated thermodynamic system is assumed to changeits con�guration from a state with energy E1 to energy E2 with probability p =exp[�(E2�E1)=kT ]. In the case E2 < E1, where p > 1, the associated probability is

179assumed to be one and the system always takes this option. This scheme of usuallytaking a downhill step while sometimes taking an uphill step has come to be knownas the Metropolis algorithm, named after the person who �rst incorporated it intonumerical calculations. The lower the temperature, the less likely is any signi�cantuphill excursion.As described in [46], one must provide the following elements to make use of theMetropolis algorithm for problems not involving thermodynamic systems:1. A description of possible system con�gurations.2. An objective function E (analog of energy) whose minimization is the goal ofthe procedure.3. A control parameter T (analog of temperature) and an annealing schedule whichdescribes how T should be lowered from high to low values.4. A generator of random changes in the con�guration; these changes are the\options" presented to the system.The simulated-annealing approach has been applied to a wide variety of com-binatorial problems, such as the traveling salesman problem of �nding the shortestcyclical itinerary for a traveling salesman who must visit each of N cities in turn,and the problem of integrated circuit design. In these cases, the free parameterstake on discrete values; the \steps" of the Monte Carlo random walk correspond topermutations in the list of cities to be visited, interchanges of circuit elements, orother discrete operations.The basic ideas behind simulated annealing are also applicable to optimizationproblems with continuous N -dimensional control spaces, e.g., �nding the global min-imum of some function f(x) in the presence of many local minima, where x is an

180N -dimensional vector. The four elements in the Metropolis procedure are now asfollows: The objective function is the value of f . The system state is the point x.The control parameter T is, as before, an analog to temperature, with an annealingschedule by which it is gradually reduced. The generator of random states, i.e., aprocedure for taking a random step from x to x+�x, remains to be de�ned.This choice of random steps becomes the principal complication in going fromthe discrete to the continuous application of simulated annealing. As described in[46], the problem is one of e�ciency: a generator of random changes is ine�cientif, when a local downhill move exists, it nevertheless almost always proposes anuphill move. In their view, a good generator should not become ine�cient in narrowvalleys; nor should it become more and more ine�cient as convergence to a minimumis approached. This seems reasonable, and one good choice for a generator appearsto be that of Vanderbilt and Louie [68], who describe a self-regulatory mechanismfor choosing the random step distribution.We have followed the method described in [46] for continuous minimization bysimulated annealing in order to obtain solutions to the problem de�ned by Equation4.8. This reference uses a modi�cation of the downhill-simplex method. The modi-�cation amounts to replacing the single point x as a description of the system stateby a general simplex 1 of n+ 1 points.The downhill-simplexmethod is based on comparing the objective-function valuesat each of the (n+ 1) vertices of the simplex and then moving the simplex towardsthe optimum point. This movement is achieved by three basic operations: re ection,contraction, and expansion. Used by itself, this method can frequently come upwith sub-optimal solutions if the objective function is not su�ciently smooth in1A set of (n + 1) equidistant points in n-dimensional space forms a regular simplex. A generalsimplex is a simplex without the property that the vertices are equally distanced.

181n-dimensional space.The Metropolis algorithm is implemented with the simplex method as follows:A positive, logarithmically distributed random variable, proportional to the temper-ature T , is added to the stored function value associated with every vertex of thesimplex, and a similar random variable is subtracted from the function value of everynew point that is tried as a replacement point, simulating thermal Brownian motion.This method always accepts a true downhill step, but sometimes accepts an uphillone. In the limit T ! 0, this algorithm reduces exactly to the downhill simplexmethod.At a �nite value of T , the simplex expands to a scale that approximates thesize of the region that can be reached at this temperature, and then samples new,approximately random, points within this region. The e�ciency with which a regionis explored is independent of its narrowness and orientation. If the temperature isreduced su�ciently slowly, it becomes likely that the simplex will shrink into thatregion containing the lowest relative minimum encountered.Consider the following unconstrained minimization problemminM(x)where x 2 <n:We will make use of the following notation:1. rnd is a function that returns a random number between 0 and 1 (non-inclusive).2. xh is the vertex of the simplex such thatM(xh)� T ln(rndh) = maxi (M(xi)� T ln(rndi))i.e. the vertex with the maximum value of the objective function plus uc-tuation. The subscript h on rnd is used to highlight the fact that its value

182is di�erent each time the function is called. Also, the uctuation quantity ispositive, since logarithms are negative on the interval (0,1).3. xs is the vertex with the second highest value of M plus associated uctuation.4. xl is the vertex with the lowest value of M plus associated uctuation.5. x0 is the centroid of all xi, except i = h, given byx0 = 1n n+1Xi=1i 6=h xi6. yi; i=1:::n+1 are the objective function values at the simplex vertices with asso-ciated uctuations: yi = y(xi) = M(xi)� T ln(rndi):(The function value at each vertex is given a random thermal- uctuation eachtime it is looked at).The three basic operations on the simplex are:re ection, where xh is replaced byxr = (1 + �)x0 � �xh � > 0;expansion, where xr is expanded in the direction along which a further improvementof the function value may be expectedxe = xr + (1� )x0 > 1;and contraction, by which we contract the simplexxc = �xh + (1 � �)x0 0 < � < 1:

183For graphical descriptions of the motion that these functions describe see [22]. Thefollowing pseudo-code describes the algorithm:while (not converged) do> Determine which vertex of the simplex has the highest (worst),> next-highest and lowest (best) value of the objective function.compute( xh, xs, xl, x0, yi; i=1:::n+1 )> Re ect the simplex from the high point.xr (1 + �)x0 � �xh> Subtract the uctuation to simulate Brownian motion.y(xr) M(xr) + T ln(rndi)> Replace simplex vertex xh by xr if possible.if (y(xr) < y(xh)) thenxh xry(xh) y(xr)> Expand simplex further if xr is a good choice.if (y(xr) < y(xl)) thenxe xr + (1 � )x0y(xe) M(xe) + T ln(rndi)if (y(xe) < y(xr)) thenxh xey(xh) y(xe)> Else contract if the re ected point is worse than the second-highest> point.else if (y(xr) > y(xs)) thenxc �xh + (1� �)x0

184y(xc) M(xc) + T ln(rndi)> Replace simplex vertex xh by xc if possible.if (y(xc) < y(xh)) thenxh xc> If the contracted point is worse than the highest point> then try contracting about the lowest point instead.else xh �xl + (1� �)x0decrease T in schedulecompute convergence criterionenddoSee [22] for suggestions regarding the construction of the initial simplex andchoices of �; � and . As with all applications of simulated annealing, success orfailure of the annealing algorithm is often determined by the choice of annealingschedule, i.e. the procedure used to reduce the temperature T in the algorithm. [46]suggests some possibilities for the schedule. The reader is referred there for moredetails. The question whether to undertake an occasional restart is also consideredin this reference.The method has several attractive features that are unique when compared toother optimization techniques. First, it is not \greedy", in the sense that it is noteasily fooled by the quick payo� achieved by falling into unfavorable local minima.Second, con�gurations tend to proceed in a logical order. Changes that cause thegreatest energy di�erences are sifted over when the control parameter T is large.These decisions become more permanent as T is lowered and the focus shifts tosmaller re�nements in the solution.

185�7:50�6:00�4:50�3:00�1:50 0:00 1:50 3:00�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) 4-stage, �(~�; �) = 0:41641st order Modi�ed Roe Scheme,M = 0:1, � = 0�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.0Figure A.1: Design-graph of multi-stage coe�cients used as an initial guess. First-order preconditioned spatial operator with M = 0:1, � = 0�.�1 = 0:1,�2 = 0:2, �3 = 0:35 and � = 1:5. � = 0:4164.Description of a typical optimization runWe will describe the solution of the optimization problem considering dampingover the entire high-frequency domain (Section 4.5) for the �rst-order four-stagescheme presented in Table 4.1.We took the following coe�cients as our initial guess: �1 = 0:1, �2 = 0:2,�3 = 0:35 and � = 1:5. Figure A.1 shows the design-graph for these coe�cients. Thecorresponding value of � was 0:4164. We actually made use of Tai's coe�cients as ourinitial guess when obtaining the results in Chapter IV. The rather arbitrary startingvalues taken here are further from the optimal solution than Tai's coe�cients, inorder to illustrate the robustness of simulated annealing.We have made use of a geometric series for our temperature schedule. We usedan initial value of 0.25 and then decremented the temperature at each iteration by afactor of 0.8. Figure A.2 shows this schedule, plotted versus iteration.

1860: 20: 40: 60:�8:0�4:00:04:0 Iterlog(T ) r = 0:8

Figure A.2: Temperature schedule that was used to solve the optimization problem.The temperature was reduced at each step by using a multiplicativefactor of 0.8.Figure A.3 shows the design-graph of the �nal solution. The value of � has beenreduced to 0:2349.Figure A.4 shows the convergence histories of the variables in the optimization(scaled by the initial guess value), as well as the value of � at each step. It can beobserved that large changes are made to the parameters in the initial steps, with thechanges gradually decreasing as the temperature decreases.Our experience indicates that the temperature schedule chosen here is fairly con-servative. The algorithm was also observed to converge quickly towards the optimalsolution when the temperature was decremented with a factor of 0:2. With thissteep schedule, the �nal value of � turned out to be slightly higher than before; thedi�erence, though, is only in the fourth signi�cant digit.The algorithm turns out to be very robust for this case. It was able to convergetowards the optimal solution with even an exponential decrease in the temperature.

187�9:00�7:50�6:00�4:50�3:00�1:50 0:00 1:50�5:25�3:75�2:25�0:750:752:253:755:25

<(z)=(z) Optimized 4-stage, �(~�; �) = 0:23861st order modi�ed Roe Scheme,M = 0:1, � = 0�1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 1.01 10Figure A.3: Design-graph of optimal multi-stage coe�cients. First-order precondi-tioned spatial operator withM = 0:1, � = 0�.�1 = 0:07876, �2 = 0:2004,�3 = 0:4241 and � = 2:63. � = 0:2349

0: 20: 40: 60:0:000:801:602:40Iter

�1=�1i�2=�2i�3=�3i�=�i�Figure A.4: Convergence histories of the parameters involved in the optimization.Also plotted are the values of � at each step.

188However, the chances of obtaining sub-optimal solutions increase as the number ofvariables in the optimization is increased. More care needs to be taken in choosing atemperature schedule then. Choosing a more aggressive temperature schedule couldalso create problems for the formulation with prescribed damping (Section 5.2): thesolution could diverge if the underlying optimization problem was solved incorrectly.The schedule that we have described worked well for all the cases that we haveconsidered (a 6-parameter optimization at most).

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189

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ABSTRACTMULTIGRID SOLUTION OF THE EULER EQUATIONS WITH LOCALPRECONDITIONINGbyJohn Francis LynnChairperson: Bram van LeerA multigrid method for solution of the steady two-dimensional Euler equations ispresented. The combination of local preconditioning with multigrid relaxation makesthe multigrid method very e�cient in obtaining steady-state solutions.The key to the success of this combination is the development of single-grid march-ing schemes with guaranteed high-frequency damping. An optimization formulationis described that may be used to obtain multi-stage schemes with superior damping;the optimization is taken over the high-frequency content in the Fourier footprintof the preconditioned spatial operator. Both standard and semi-coarsened multi-grid have been considered, requiring optimization over di�erent frequency domains.The optimization problem has been solved by the method of simulated annealingtogether with the downhill-simplex method. Tables of multi-stage coe�cients havebeen presented that are based on the solution to this optimization problem.

1It is shown that the combination of local preconditioning and multi-stage time-stepping can produce relaxation schemes that boast strong high-frequency dampingfor the entire range of ow angles, Mach numbers, cell aspect-ratios and (for Navier-Stokes operators) cell-Reynolds numbers. Such schemes are ideally suited for useas relaxation schemes in a multigrid framework, particularly if semi-coarsening isused. In addition they are superior relaxation schemes if only a single grid is used, incomparison to other explicit marching schemes with or without local preconditioning.The preconditioning already accelerates the convergence to the steady state and thehigh-frequency damping provides robustness.Multigrid Euler solutions on structured meshes are presented as test cases. Thesenumerical studies indicate that multigrid speed-ups of a factor of 3-4 may be ob-tained when local preconditioning is used. Studies also indicate that explicit residual-smoothing can further improve convergence rates by upto 25%, as well as improvingrobustness, with only a minimal increase in the computational e�ort required perupdate.The extension to Navier-Stokes operators and three space dimensions, and theimplementation on unstructured meshes are also brie y considered.