Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration

TL;DR

This paper introduces a novel local subcell monolithic DG/FV scheme on unstructured grids that enhances robustness and entropy stability while maintaining high accuracy through convex blending of schemes.

Contribution

It develops a new monolithic DG/FV method with convex blending on subcells, improving robustness and entropy stability without sacrificing high-order accuracy.

Findings

The scheme effectively preserves convex properties and entropy stability.

Numerical results demonstrate high accuracy and robustness.

Optimal blending coefficients are identified for desired properties.

Abstract

This article aims at presenting a new local subcell monolithic Discontinuous-Galerkin/Finite-Volume (DG/FV) convex property preserving scheme solving system of conservation laws on 2D unstructured grids. 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, a convex blending of high-order DG and first-order FV scheme will be locally performed, at the subcell scale, where it is needed. To this end, by means of the theory developed in our previous work, we first recall that it is possible to rewrite DG scheme as a subcell FV scheme on a subgrid provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, the monolithic DG/FV method will be defined as following: each face of each subcell will be assigned with two fluxes, a 1st-order FV one and a high-order reconstructed one, that in the end will be blended in a convex way. The goal is then to determine, through analysis, optimal blending coefficients to achieve the desire properties. Numerical results on various type problems will be presented to assess the very good performance of the design method. A particular emphasis will be put on entropy consideration. By means of this subcell monolithic framework, we will attempt to address the following questions: is this possible through this monolithic framework to ensure any entropy stability? what do we mean by entropy stability? What is the cost of such constraints? Is this absolutely needed while aiming for high-order accuracy?