Extension Of The High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme To Solve Time Dependent Diffusion Equations

Abstract

In this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common “variational crime” problem when traditional DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods.

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

“…Note that the treatment of diffusion terms in traditional semi-discrete DG methods are non-trivial because of the so-called "variational crime". More test cases on diffusion problems have been presented in [7].…”

Accuracy Enhancement of a Riemann-Solver-Free Spacetime Discontinuous Galerkin Method via Constrained Least Square Reconstruction

“…[1][2][3] The method was inspired by the well-known Conservation Element/Solution Element (CE/SE) method 4 and the discontinuous Galerkin (DG) method. 5 The method adopts the concepts of using staggered spacetime meshes to enforce spacetime flux conservation (cf.…”