\textit{A Posteriori} local subcell correction of high-order discontinuous Galerkin scheme for conservation laws on two-dimensional unstructured grids

TL;DR

This paper extends a posteriori local subcell correction techniques for high-order discontinuous Galerkin schemes to two-dimensional unstructured grids, improving solution robustness, positivity, and shock resolution while maintaining high accuracy.

Contribution

It introduces a 2D unstructured grid extension of the local subcell correction for DG schemes, enhancing stability and physical admissibility without sacrificing accuracy.

Findings

Ensures positivity preservation of solutions.

Achieves sharp shock representation with low oscillations.

Maintains high-order accuracy and local conservation.

Abstract

We present the two-dimensional unstructured grids extension of the a posteriori local subcell correction of discontinuous Galerkin (DG) schemes introduced in [52]. 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 local 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, etc). Numerical results on various type problems and test cases will be presented to assess the very good performance of the design correction algorithm.