A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes

In this paper, we consider numerical approximations of diffusion terms for finite volume as well as discontinuous Galerkin schemes. Both classes of numerical schemes are quite successful for advection equations capturing strong gradients or even discontinuities, because they allow their approximate solutions to be discontinuous at the grid cell interfaces. But, this property may lead to inconsistencies with a proper definition of a diffusion flux. Starting with the finite volume formulation, we propose a numerical diffusion flux which is based on the exact solution of the diffusion equation with piecewise polynomial initial data. This flux may also be used by discontinuous Galerkin schemes and gives a physical motivation for the Symmetric Interior Penalty discontinuous Galerkin scheme. The flux proposed leads to a one-step finite volume or discontinuous Galerkin scheme for diffusion, which is arbitrary order accurate simultaneously in space and time. This strategy is extended to define suitable numerical fluxes for nonlinear diffusion problems.

[1]  Bram Van Leer,et al.  Discontinuous Galerkin for Diffusion , 2005 .

[2]  Mengping Zhang,et al.  AN ANALYSIS OF THREE DIFFERENT FORMULATIONS OF THE DISCONTINUOUS GALERKIN METHOD FOR DIFFUSION EQUATIONS , 2003 .

[3]  Suzanne L. Weekes.  The Travelling Wave Scheme for The Navier--Stokes Equations , 1998 .

[4]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[5]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[6]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[7]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[8]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[10]  P. Lax,et al.  Systems of conservation laws , 1960 .

[11]  Claus-Dieter Munz,et al.  Lax–Wendroff-type schemes of arbitrary order in several space dimensions , 2006 .

[12]  Eleuterio F. Toro,et al.  Towards Very High Order Godunov Schemes , 2001 .

[13]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .

[14]  Eleuterio F. Toro,et al.  Derivative Riemann solvers for systems of conservation laws and ADER methods , 2006, J. Comput. Phys..

[15]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[16]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[17]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[18]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[19]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[20]  Eduard Harabetian.  A numerical method for viscous perturbations of hyperbolic conservation laws , 1990 .

[21]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[22]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[23]  Michael Dumbser,et al.  Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes , 2006, J. Sci. Comput..

[24]  D. Arnold,et al.  Discontinuous Galerkin Methods for Elliptic Problems , 2000 .

[25]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

[26]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .