A Posteriori Local Subcell Correction of High-Order Discontinuous Galerkin Scheme for Conservation Laws on Two-Dimensional Unstructured Grids

.In this paper, we present the two-dimensional unstructured grids extension of the a posteriori local subcell correction of discontinuous Galerkin (DG) schemes introduced in F. Vilar, J. Comput. Phys., 387 (2018), pp. 245–279. The technique is based on the reformulation of the DG scheme as a finite-volume (FV)-like method through the definition of some specific numerical fluxes referred to as reconstructed fluxes. A 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 of 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 conservativity. 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. For a subcell detected as admissible, we keep the high-order DG reconstructed flux which allows us to retain the very highly 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 introduced in F. Vilar, J. Comput. Phys., 387 (2018), pp. 245–279. Numerical results on various type of problems and test cases will be presented to assess the very good performance of the designed correction algorithm.Keywordsa posteriori correctionsubcell correctionarbitrary high-orderDG subcell FV formulationpositivity-preserving schemehyperbolic conservation lawssubcell conservative schemeMSC codes65N1235L6765M0865N30