Extension of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme to Solve Time Dependent Diffusion Equations (Communications in Computational Physics, 2012)
In this paper, the high-order space-time discontinuous Galerkin cell vertexscheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of thediffusive 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 thediffusive flux are evaluated using continuous information across the cell interface. Asa result, the resulting formulation with diffusive fluxes present is still consistent anddoes not need any extra ad hoc techniques to cure the common "variational crime" problem when traditional DG methods are applied to diffusion problems. For thisreason, DG-CVS is conceptually simpler than other existing DG-typed methods. Thenumerical tests demonstrate that the convergence order based on the L2-norm is optimal, i.e. O(hp+1) for the solution and O(hp) for the solution gradients, when thebasis polynomials are of odd degrees. For even-degree polynomials, the convergenceorder is sub-optimal for the solution and optimal for the solution gradients. The sameodd-even behaviour can also be seen in some other DG-typed methods.