A POSTERIORI LOCAL SUBCELL CORRECTION OF HIGH-ORDER DISCONTINUOUS GALERKIN SCHEME FOR CONSERVATION LAWS ON TWO-DIMENSIONAL UNSTRUCTURED GRIDS
Résumé
In this paper, we present the two-dimensional unstructured grids extension of the a posteriori local subcell correction (APLSC) of discontinuous Galerkin (DG) schemes introduced in our previous article. The technique is based on the reformulation of DG scheme as a finite volume (FV) like method through the definition of some specific numerical fluxes referred to as reconstructed fluxes. High-order DG numerical scheme combined with this new local subcell correction technique ensures positivity preservation of the solution, along with a low oscillatory and sharp shocks representation. The main idea of this correction procedure is to retain as much as possible the high accuracy and the very precise subcell resolution of DG schemes, while ensuring the robustness and stability of the numerical method, as well as preserving physical admissibility of the solution. Consequently, an a posteriori correction will only be applied locally at the subcell scale where it is needed, but still ensuring the scheme conservation. Practically, at each time step, we compute a DG candidate solution and check if this solution is admissible (for instance positive, non-oscillating,. . .). If it is the case, we go further in time. Otherwise, we return to the previous time step and correct locally, at the subcell scale, the numerical solution. To this end, each cell is subdivided into subcells. Then, if the solution is locally detected as bad, we substitute the DG reconstructed flux on the subcell boundaries by a robust first-order numerical flux. And for subcell detected as admissible, we keep the high-order DG reconstructed flux which allows us to retain the very high accurate resolution and conservation of the DG scheme. As a consequence, only the solution inside troubled subcells and its first neighbors will have to be recomputed, elsewhere the solution remains unchanged. Another technique blending in a convex combination fashion DG reconstructed fluxes and first-order FV fluxes for admissible subcells in the vicinity of troubled areas will also be presented and prove to improve results in comparison to the original algorithm. Numerical results on various type problems and test cases will be presented to assess the very good performance of the design correction algorithm.
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