A limiter-based approach to construct high-order fully-discrete entropy stable explicit DG schemes for hyperbolic conservation laws

TL;DR

This paper introduces a new class of high-order explicit DG schemes that are entropy stable, using a limiter-based approach to enforce multiple entropy inequalities without altering the spatial discretization.

Contribution

The methodology enables simultaneous enforcement of multiple entropy inequalities in high-order DG schemes through a simple post-processing limiter, maintaining high-order accuracy.

Findings

Numerical tests show optimal accuracy and robustness.

The scheme enforces multiple entropy conditions.

Theoretical proofs confirm high-order accuracy and entropy stability.

Abstract

This paper presents a class of novel high-order fully-discrete entropy stable (ES) discontinuous Galerkin (DG) schemes with explicit time discretization. The proposed methodology exploits a critical observation from [4] that the cell averages of classical DG solutions with forward Euler time stepping satisfy an ``entropy-stable-like'' property. Building on this result, fully-discrete entropy stability is rigorously enforced through a simple Zhang--Shu-type scaling limiter [45] applied as a post-processing step, without modifying the underlying spatial discretization. Furthermore, the proposed methodology can simultaneously enforce multiple cell entropy inequalities, a capability unavailable in existing ES DG schemes. High-order accuracy in time is achieved by using strong-stability-preserving (SSP) multistep methods. Theoretically, we prove that the proposed scheme indeed maintains high-order accuracy and establish a Lax--Wendroff-type theorem guaranteeing that the limit of the numerical solutions, if it exists, satisfies the desired entropy inequality. Extensive numerical tests for scalar equations and systems, including the nonconvex Buckley--Leverett problem and extreme examples of Euler equations, demonstrate optimal accuracy, enforcement of multiple entropy conditions, and strong robustness.