A posteriori local subcell correction of DG schemes through Finite Volume reformulation on unstructured grids

  • Vilar, François (IMAG, Université de Montpellier)

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This talk aims at presenting the 2D version of the A Posteriori Local Subcell Correction (APLSC) for discontinuous Galerkin (DG) schemes recently introduced in our previous paper. This is known that DG method needs some sort of nonlinear limiting to avoid spurious oscillations or nonlinear instabilities which may lead to the crash of the code. The main idea motivating the present work is to improve the robustness of DG schemes, while preserving as much as possible its high accuracy and very precise subcell resolution. To do so, an a posteriori correction will only be applied locally at the subcell scale where it is needed. To this end, we first prove that it is possible to rewrite DG scheme as a subcell Finite Volume scheme provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, 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, this solution. Practically, if the solution on a subcell has been detected as bad, we substitute the DG reconstructed flux on its boundary by a robust first-order numerical flux. And for subcell detected as admissible, we keep the high-order reconstructed flux to retain the accurate resolution of DG scheme. Numerical results on various type problems will be presented to assess the very good performance of the design correction.