High-Order Methods Including Discontinuous Galerkin by Reconstructions on Triangular Meshes

The flux reconstruction (or correction procedure using reconstruction) approach provides a simple and economical framework to derive high-order numerical schemes for conservation laws. It employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure based on the concept of reconstruction. In addition to resulting in new schemes, the approach unifies several existing methods: with appropriate choices of correction terms, it recovers discontinuous Galerkin (DG), spectral volume, and spectral difference (or staggered-grid). The flux reconstruction versions are also generally simpler and more economical than the original versions. The reconstruction framework is extended to the case of a triangular mesh here. A DG algorithm using standard DG tools and the differential form of the equation with no quadratures is included. The current approach can be incorporated into an existing DG code with relative ease. Keywords. Discontinuous Galerkin methods, high-order methods, conservation laws.

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