High order entropy preserving ADER scheme

TL;DR

This paper introduces a novel fully discrete ADER-DG scheme that preserves entropy at machine precision, combining entropy correction in space and a relaxation-based time discretization, verified through numerical simulations.

Contribution

It is the first to develop a provably entropy-preserving fully discrete ADER-DG scheme using combined spatial entropy correction and relaxation-based time discretization.

Findings

The scheme preserves entropy to machine precision.

Numerical simulations confirm theoretical entropy preservation.

First construction of a fully discrete entropy-preserving ADER-DG method.

Abstract

In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machine precision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations.