High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting

TL;DR

This paper introduces an entropy-stable extension of subcell limiting strategies for discontinuous Galerkin spectral element methods, ensuring semi-discrete cell entropy inequality while maintaining high-order accuracy.

Contribution

It formulates limiting factors as an optimization problem solved efficiently, extending subcell limiting to preserve convex constraints and improve stability.

Findings

Preserves high-order accuracy for smooth solutions

Satisfies semi-discrete cell entropy inequality

Efficient greedy algorithm for optimization

Abstract

Subcell limiting strategies for discontinuous Galerkin spectral element methods do not provably satisfy a semi-discrete cell entropy inequality. In this work, we introduce an extension to the subcell and monolithic convex limiting strategies that satisfies the semi-discrete cell entropy inequality by formulating the limiting factors as solutions to an optimization problem. The optimization problem is efficiently solved using a deterministic greedy algorithm. We also discuss the extension of the proposed subcell limiting strategy to preserve general convex constraints. Numerical experiments confirm that the proposed limiting strategy preserves high-order accuracy for smooth solutions and satisfies the cell entropy inequality.