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Index

Abel Prize, 115Absorbing boundary conditions, 216Adams, 61Adams methods, 61Adams–Moulton methods, 63, 67ADI, see Alternating-direction implicit methodAdvancing front method, 175Algebraic multigrid methods, 194ALGOL, 139Alternating-direction implicit (ADI) method,

106Amplification factor, 65, 115Archimedes, 5, 10, 14Arnoldi, 131Arnoldi method, 131Artificial viscosity, 114Assembly language, 139A-stability, 96, 98

Bôcher, 55Babylonian collection, 6Babylonian method, 8Backward differentiation methods, 97, 143Backwards error analysis, 127Bashforth, 63Beltrami, 200Berenger, 219BESK, 93Birkhoff, 126Björck, 95Black, 228Black–Scholes equation, 229Brandt, 193Bulirsch, 146

Burgers’ equation, 112Butcher, 144Butcher tableau, 144

C++, 140CDC 6600, 134CDC 7600, 134CDC STAR 100, 134CFL-condition, 81CFL-number, 81Characteristic equation, 94Chebyshev, 50Chebyshev polynomial, 50Chinese mathematics, 15Cholesky, 43Cholesky factorization, 43Clairaut, 32Cloud computing, 138Collocation methods, 164, 186Compiler, 139Computational chemistry, 237Computational engineering, 238Computational physics, 236Conjugate direction method, 205Conservative form, 117Cosine-transform, 57Cotes, 33, 34Courant, 73, 79, 114Courant Institute, 73Courant number, 81Crank, 100Crank–Nicolson method, 76, 105Cray X-MP, 135Cray Y-MP, 135

© Springer International Publishing AG, part of Springer Nature 2018B. Gustafsson, Scientific Computing, Texts in Computational Scienceand Engineering 17, https://doi.org/10.1007/978-3-319-69847-2

249

250 Index

Cray-1, 134Cray-2, 135Curtiss, 96

Dahlquist, 95, 96Dahlquist barrier, 98d’Alembert’s formula, 80Damped Jacobi method, 189Dantzig, 85Daubechies, 159de Boor, 155Debye, 88Deferred correction methods, 214Delaunay triangulation, 174Determinant condition, 152Difference methods ODE, 94Difference methods PDE, 99Differential–algebraic equations, 99Dirichlet, 52Dirichlet boundary conditions, 106Discontinuous Galerkin methods, 171Discrete Fourier series, 101Discrete Fourier transform, 57, 184Dissipative methods, 119Domain decomposition, 210Domain of dependence, 80Douglas Jr., 106Dryja, 215

Eckert, 93EDVAC, 93EISPACK, 140Engquist, 197, 216ENIAC, 92ENO methods, 231Equivalence theorem, 109Euclid, 10Euler, 60Euler backward method, 60, 97Euler equation, 167Euler method, 58, 96Explicit methods, 59

Fast Fourier transform, 175Fedorenko, 189FEM, 72, 165Fields Medal, 142Financial mathematics, 228Finite element methods, 72, 165Finite volume method, 118Fixed point method, 19

Fluid dynamics, 111FMM, 195FORTRAN, 139Fourier, 51

analysis, 99coefficients, 53method, 185series, 51transform, 56

Fourier–Motzkin algorithm, 83Friedrichs, 79, 114Fujitsu, 135

Galerkin, 70Galerkin method, 71, 171Gauss, 35, 40Gaussian elimination, 42Gauss–Lobatto points, 50Gauss–Seidel method, 45Gear, 145Generalized eigenvalue, 152Generalized Minimal Residual Method, 208Gibbs, 55Gibbs phenomenon, 55, 182Givens, 130Givens method, 124GKS-theory, 152GMRES, 208Godunov, 117Godunov method, 118Godunov–Ryabenkii condition, 150Goldstine, 93Golub, 202Gottlieb, 186Gragg, 146Gram, 122Gram–Schmidt orthogonalization, 122Gravity waves, 77, 98Greengard, 195Green’s formula, 73Gregory, 48Grid generation, 174

Haar, 158Haar wavelet, 160Hackbusch, 193Heat equation, 100Henrici, 146Hermite, 28Heron, 8Heron’s method, 8Hessenberg form, 131

Index 251

Hestenes, 203Heterogeneous multiscale methods, 197Heun, 64High Performance Computer (HPC), 136Hirschfelder, 96Hitachi SR2201, 136HMM, 197Horner’s scheme, 16Householder transformation, 123HPC, see High Performance ComputerHui, 16, 42Huygens, 65Hyperbolic problem, 99

IBM 704, 139IBM 7090, 134ILLIAC-IV, 135Implicit methods, 59, 105Initialization, 78Iterative methods, 7, 125

Jacobi, 24Jacobian, 24Jacobi method, 45Jacobi rotation, 129Jameson, 238Java, 140Jordan, 200

Kantorovich, 85Kepler, 17, 19Kepler’s equation, 19Kopernikus, 17Kreiss, 109, 150Krylov, 206Krylov space methods, 131, 203, 205k-step method, 62Kutta, 64

Lagrange, 27, 88function, 88interpolation, 28multiplier method, 88

Lanczos, 186Lanczos method, 131Laplace operator, 164Lax, 108, 113Lax–Friedrichs scheme, 115Le Verrier, 61Leap-frog scheme, 64, 76, 102Least squares method, 36

Legendre, 37Legendre polynomials, 40, 49Leibniz, 22, 87Level set methods, 229Lewy, 79Linearization, 8Linear programming, 85LINPACK, 137, 140Lions, Jacques–Louis, 142Lions, Pierre–Louis, 142Longley, 18LR algorithm, 130LU decomposition, 43

Machine code, 139MacLaurin, 48MacLaurin series, 48Macroscale, 197Majda, 216Manhattan project, 90Mark I, 91Mass matrix, 171MATLAB, 140Matrix theorem, 109Mauchly, 93Maxwell equations, 219Method of lines, 147Methods of Fluxions, 46Microprocessor, 135Microscale, 197Millennium Prize, 142Mises, 129Moler, 140Monte Carlo-methods, 221Moore, 200Moore–Penrose pseudo-inverse, 200Motzkin, 82Moulton, 63Multigrid methods, 188Multipole methods, 195Multiscale methods, 188Multistep methods, 62, 98

NASA, 94Navier–Stokes equations, 142Newton, 20, 87Newton–Cotes formulas, 34Newton methods, 20Newton–Raphson method, 23, 87Nicolson, 100Nobel Memorial Prize, 228Nobel Prize, 69, 87, 88, 180

252 Index

Nonconforming elements, 173Nonlinear optimization, 87Nonlinear problems, 141Nonreflecting boundary conditions, 221Numerical domain of dependence, 80Numerical wind tunnel, 136

ODE, 58, 94, 143Open boundaries, 215Operator splitting, 105Optimization, 81Orthogonal functions, 39Orthogonalization, 121Orthogonal polynomials, 39, 48Ossendrijver, 8

Parabolic problems, 100Parallel computer, 136Parallel vector computer, 135Parasitic solution, 78Partial differential equations (PDE), 68, 99,

148PDE, see Partial differential equationsPeaceman, 106Penrose, 200Pereyra, 214Perfectly Matched Layer (PML), 220Personal computer, 138Piazzi, 36Picard theorem, 141Piecewise polynomials, 154Pivoting, 43PML, see Perfectly Matched LayerPoincaré, 65Poisson equation, 72Pollaczek–Geiringer, 129Pope, 21Power method, 129Preconditioning, 208Programming, 139Pseudo-differential operator, 217Pseudo-inverse, 200Pseudo-random numbers, 227Pythagoras, 6Pythagorean theorem, 6

QR-algorithm, 131

Rachford, 106Radial basis functions, 163Random numbers, 224

Random walk, 225Raphson, 22Rayleigh–Ritz method, 69Recursion formula, 101Richardson, 66, 67, 75Richardson extrapolation, 65, 146Richtmyer, 108Riemann, 5

function, 5hypothesis, 5problem, 117

Ritz, 69Ritz–Galerkin method, 69, 71Rokhlin, 195Runge, 30, 64

function, 30phenomenon, 31

Runge–Kutta methods, 64, 144Rutishauser, 94, 130

Saigey, 66Scarborough, 18Schmidt, 122Schoenberg, 31, 154Scholes, 228Schur complement method, 213Schwarz, 210Schwarz alternating method, 211Scientific computing, 92Series expansion, 46Shepp–Logan phantom, 182Shock capturing method, 113Shock fitting methods, 119Shocks, 111Shoujing, 16Similarity transformation, 128Simplex method, 85Simpson, 34Simpson’s rule, 34Sine-transform, 57Singular value decomposition (SVD), 199SOR method, 126Spectral element methods, 187Splines, 154Stability ODE, 94Stability PDE, 108Steepest descent, 87, 203Stiefel, 203Stiff ODE, 96Stoer, 146Sunway TaihuLight, 136SVD, see Singular value decompositionSymbolic processing, 140

Index 253

Taylor, 47Taylor series, 47Tianhe-2, 137Time-dependent PDE, 111Trapezoidal method, 34, 67, 98Trigonometric interpolation, 31, 57Turing, 89Turing Award, 138Turing’s machine, 90Turner, 166Two-grid methods, 192Two-step methods, 101

Ulam, 221Upwind scheme, 118

Variational problem, 69Vector processor, 134von Neumann, 90von Neumann condition, 99, 104von Seidel, 45

Wavelets, 158Weather prediction, 77, 102Weierstrass, 30Weierstrass theorem, 30Weinan, 197Well posed, 108Welsch, 35Wendroff, 116Whittaker, 18Widlund, 215Wilbraham, 55Wilkinson, 127

Xerox Alto, 138

Yeh, 16Young, 126

Zienkiewicz, 169

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117. T. Sakurai, S. Zhang, T. Imamura, Y. Yusaku, K. Yoshinobu, H. Takeo (eds.), Eigenvalue Problems:Algorithms, Software and Applications, in Petascale Computing. EPASA 2015, Tsukuba, Japan,September 2015.

118. T. Richter (ed.), Fluid-structure Interactions. Models, Analysis and Finite Elements.

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122. A. Gerisch, R. Penta, J. Lang (eds.), Multiscale Models in Mechano and Tumor Biology. Modeling,Homogenization, and Applications.

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