Commun. Comput. Phys., 11 (2012), pp. 1503-1524.


Extension of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme to Solve Time Dependent Diffusion Equations

Shuangzhang Tu 1*, Gordon W. Skelton 1, Qing Pang 1

1 Department of Computer Engineering, Jackson State University, Jackson, MS 39217, USA.

Received 5 August 2010; Accepted (in revised version) 9 June 2011
Available online 23 December 2011
doi:10.4208/cicp.050810.090611a

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. The numerical tests demonstrate that the convergence order based on the $L_2$-norm is optimal, i.e. $O(h^{p+1})$ for the solution and $O(h^p)$ for the solution gradients, when the basis polynomials are of odd degrees. For even-degree polynomials, the convergence order is sub-optimal for the solution and optimal for the solution gradients. The same odd-even behaviour can also be seen in some other DG-typed methods.

AMS subject classifications: 65M99, 76M25

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Key words: High-order method, space-time method, discontinuous Galerkin (DG) method, cell-vertex scheme (CVS), diffusion equations.

*Corresponding author.
Email: shuangzhang.tu@jsums.edu (S. Tu), gordon.skelton@jsums.edu (G. Skelton), qing.pang@jsums.edu (Q. Pang)
 

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