Equivalence between the Energy Stable Flux Reconstruction and Filtered Discontinuous Galerkin Schemes: Numerical Verification

摘要

The aim of this paper is to numerically demonstrate the equivalence between filtered Discontinuous Galerkin (DG) schemes and the Energy Stable Flux Reconstruction (ESFR) schemes in the general context of higher dimensional curvilinear element formulations. We first present the theory, as outlined in our previous work, demonstrating that all ESFR scheme correction fields can be interpreted as modally filtered DG correction fields in the strong form of the schemes, and further, that ESFR schemes can be interpreted as a DG scheme in weak form where discontinuous edge flux is substituted for numerical edge flux correction. The theoretical derivations are then verified with numerical results obtained for a 3D Euler test case with curved geometry on both tetrahedral and hexahedral meshes. Given the current choice of high-order DG-type schemes and the question as to which might be best to use for a specific application, the main significance of this work is the bridge that it provides between them. Clearly outlining the similarities between the schemes results in the important conclusion that it is always less efficient to use ESFR schemes, as opposed to the weak DG scheme, when solving problems implicitly.