Abstract
In this paper, we present the two-dimensional unstructured grids extension of the aposteriori 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 DGscheme as a finite-volume (FV)-like method through the definition of some specific numerical fluxesreferred to as reconstructed fluxes. A high-order DG numerical scheme combined with this newlocal subcell correction technique ensures positivity preservation of the solution, along with a lowoscillatory and sharp shocks representation. The main idea of this correction procedure is to retainas 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 physicaladmissibility of the solution. Consequently, an a posteriori correction will only be applied locally atthe subcell scale where it is needed, but still ensuring the scheme conservativity. Practically, at eachtime step, we compute a DG candidate solution and check if this solution is admissible (for instancepositive, non-oscillating,. . .). If it is the case, we go further in time. Otherwise, we return to theprevious time step and correct locally, at the subcell scale, the numerical solution. To this end, eachcell is subdivided into subcells. Then, if the solution is locally detected as bad, we substitute theDG reconstructed flux on the subcell boundaries by a robust first-order numerical flux. For a subcelldetected as admissible, we keep the high-order DG reconstructed flux which allows us to retainthe very highly accurate resolution and conservation of the DG scheme. As a consequence, onlythe 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 DGreconstructed fluxes and first-order FV fluxes for admissible subcells in the vicinity of troubledareas will also be presented and prove to improve results in comparison to the original algorithmintroduced in [F. Vilar,J. Comput. Phys., 387 (2018), pp. 245--279]. Numerical results on varioustype of problems and test cases will be presented to assess the very good performance of the designedcorrection algorithm.
Vilar, François