@Article{CiCP-11-5,
author = {},
title = {Extension of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme to Solve Time Dependent Diffusion Equations},
journal = {Communications in Computational Physics},
year = {2012},
volume = {11},
number = {5},
pages = {1503--1524},
abstract = {<p style="text-align: justify;">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 &quot;variational crime&quot; 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<sup>2</sup>-norm is optimal, i.e. O(h<sup><span style="text-align: justify;"><span style="text-align: justify;">p+1</span></span></sup>) for the solution and <span style="text-align: justify;">O</span>(h<sup>p</sup>) 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.050810.090611a},
url = {https://global-sci.com/article/80756/extension-of-the-high-order-space-time-discontinuous-galerkin-cell-vertex-scheme-to-solve-time-dependent-diffusion-equations}
}