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Journal of Computational Physics 153, 353377 (1999)Article ID jcph.1999.6281, available online at http://www.idealibrary.com on

Diffusion Characteristics of Finite Volumeand Fluctuation Splitting Schemes

William A. Wood and William L. Kleb

NASA Langley Research Center, Hampton, Virginia 23681

E-mail:

w.a.wood@larc.nasa.gov; w.l.kleb@larc.nasa.gov

Received November 13, 1998; revised March 29, 1999

The diffusive characteristics of two upwind schemes, multi-dimensional uctu-ation splitting and locally one-dimensional nite volume, are compared for scalaradvectiondiffusion problems. Algorithms for the two schemes are developed fornode-based data representation on median-dual meshes associated with unstructuredtriangulations in two spatial dimensions. Four model equations are considered: lin-ear advection, non-linear advection, diffusion, and advectiondiffusion, with caseschosen to mimic features present in compressible gas dynamics. Modular coding isemployed to isolate the effects of the two approaches for upwind ux evaluation,allowing for head-to-head accuracy and efciency comparisons. Both the stability of compressive limiters and the amount of articial diffusion generated by the schemesare found to be grid-orientation dependent, with the uctuation splitting schemeproducing less articial diffusion than the nite volume scheme. Convergence ratesare compared for an advectiondiffusion problem, with a speedup of 2.5 seen foructuation splitting versus nite volume when solved on the same mesh. However,accurate solutions to problems with small diffusion coefcients can be achieved oncoarser meshes using uctuation splitting rather than nite volume, so that whencomparing convergence rates to reach a given accuracy, uctuation splitting showsa speedup of 29 over nite volume for the test problem. c 1999 Academic Press

Key Words: nite volume; uctuation splitting; articial dissipation.

INTRODUCTION

Upwind discretizations for advection equations typically introduce articial numericaldissipation into the solution. When combined advectiondiffusion problems are considered,this dissipation introduced in the discretization of the advection terms should be less than thetrue physical diffusion. Here, the diffusive charactersitics of upwind advection schemes are

investigated on unstructured triangulations, and their performance in resolving solutions tocombined advectiondiffusion problems, with small diffusion coefcients, is quantitativelyassessed.

353

0021-9991/99 $30.00Copyright c 1999 by Academic Press

All rights of reproduction in any form reserved.

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354 WOOD AND KLEB

Two node-based, median-dual methods for modeling convective uxes are considered.The rst is a traditional locally one-dimensional approximate Riemann solver nite volume(FV) scheme [1]. Locally one-dimensional schemes applied on multidimensional domainsare known to introduce excess dissipation when discontinuities are not aligned with themesh [2].

The second method is the narrow non-linear [3] uctuation splitting (FS) scheme, alsoreferred to in the literature as a residual distribution scheme. FS has a more compact stencilthan FV for second-order formulations and exhibits zero cross diffusion in a grid-alignedcondition. Both of these attributes should lead to less articial dissipation as compared withFV.

The sensitivity of FS and FV to grid orientation and resulting production of cross diffusion

is investigated in the present report. The use of compressive limiter functions is also testedwith both algorithms. Local timesteps based on positivity arguments are implemented forboth rst- and second-order discretizations of the implicit matrix.

Formulation of FS schemes for diffusion problems is a recent research area [4, 5]. Thepresent study seeks to quantify therelativemeritsof using a low-diffusionadvectionoperatorto resolve advectiondiffusion problems with small diffusion coefcients. Lessons learnedon these problems will guide the development of computer codes for solving compressibleviscous uid dynamic problems. A similar approach for central difference schemes withexplicit numerical dissipation has recently been taken by Efraimsson [6].

The model problems considered are linear advection, non-linear advection, linear dif-fusion, and linear advectiondiffusion. These problems are intended to mimic behaviorexpected from applications to gas dynamics such as supersonic, inviscid, perfect-gas ow(non-linear advection), and high-Reynolds-number shear ows (advectiondiffusion).

GOVERNING EQUATIONS

The non-linear advectiondiffusion equation,

u t + F = ( u ), (1)

is cast as a hyperbolic conservation law, to which steady-state solutions are sought. Hereu is the dependent variable, F = F ( x, y, u ) is the convective ux function, and is thediffusion coefcient.

Finite Volume

In FV form, using the divergence theorem Eq. (1) becomes

u t d = ( F u ) n d , (2)where is the median dual about node i , is the boundary of , and n is the outward

unit normal to control cell. Using mass lumping to the nodes, similar to an explicit niteelement treatment [7], the temporal evolution is evaluated on a time-invariant mesh as

u t d = S i u i t S i u t + i u t i , (3)where S i is the median-dual area about node i and is the time step.

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DIFFUSION CHARACTERISTICS OF ADVECTION SCHEMES 355

The discretization of the convective ux, F , is constructed following Barths [1] imple-mentation of the upwind, locally one-dimensional, approximate Riemann solver of Roe[8],

F n d faces 12 ( F in + F out) n , (4)where the articial dissipation, , provides the upwinding,

= 12

| An x + Bn y|(u out u in), (5)

with n = n x + n y ( and are the Cartesian unit vectors). Out and in refer to stateson the outside and inside of at the face. A and B are the ux Jacobians in the and directions, respectively,

A = F (1)

u, B =

F (2)

u, (6)

and ( A, B) represent their conservative linearizations at the cell face [8].Piecewise linear reconstruction from the nodal unknowns to the cell faces as

u face = u i + u r (7)

provides second-order spatial accuracy in smoothly varying regions of the solution. r is theposition vector from the node to the face. Median-dual gradients of the dependent variable,

u , are obtained from the unweighted least squares procedure outlined by Barth. FollowingBruner and Walters [9], the limiter function, , is supplied an argument equal to half theargument Barth uses, namely,

= umin/max u i2( u r )min/max

, (8)

where umin/max is the minimum (resp. maximum) of u i and all distance-one neighbors. The

more restrictive constraint from using either the maximum or minimum is used to set thelimiter value.

In casting the limiter argument in this form, Bruner equates the Barth limiter withSuperbee, for a limiter argument less than or equal to one. The Barth limiting is non-symmetric, taking the form

pq =

0 pq 0

2 pq if 0