Volume Term Adaptivity for Discontinuous Galerkin Schemes

arXiv (Cornell University)

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.

Authors 6

1. Daniel Doehring Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics , RWTH Aachen University , Germany

2. Jesse Chan

   The University of Texas at Austin

   Affiliation as printed

   Oden Institute for Computational Science and Engineering , Aerospace Engineering & Engineering Mechanics , University of Texas at Austin , USA

3. Hendrik Ranocha

   Johannes Gutenberg University Mainz

   Affiliation as printed

   Institute of Mathematics , Johannes Gutenberg University Mainz , Germany

4. Michael Schlottke-Lakemper

   University of Augsburg

   Affiliation as printed

   High-Performance Computing , Center for Advanced Analytics and Predictive Sciences , University of Augsburg , Germany

5. Manuel Torrilhon Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics , RWTH Aachen University , Germany

6. Gregor J. Gassner

   University of Cologne

   Affiliation as printed

   Department of Mathematics and Computer Science , Center for Data and Simulation Science , University of Cologne , Germany