Volume Term Adaptivity for Discontinuous Galerkin Schemes

TL;DR

This paper introduces v-adaptivity for high-order discontinuous Galerkin schemes, dynamically choosing volume term discretizations at each Runge-Kutta stage to enhance robustness, efficiency, and accuracy in solving time-dependent PDEs.

Contribution

A novel v-adaptivity approach for DG schemes that adaptively selects volume discretizations based on entropy considerations to improve performance.

Findings

V-adaptive DG schemes maintain accuracy and stability.

The approach enhances robustness and efficiency.

Effective in 2D and 3D compressible flow problems.

Abstract

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.