Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations

TL;DR

This paper presents a new optimization-based approach using splitting methods to enforce invariant domain constraints in high order numerical schemes for gas dynamics, improving robustness and accuracy.

Contribution

It introduces explicit projection formulations and applies classical splitting methods to develop invariant-domain-preserving limiters for high order schemes.

Findings

Successfully applied to high order discontinuous Galerkin schemes

Validated robustness and performance on demanding benchmarks

Demonstrated effectiveness of $$- and $$-norm minimization limiters

Abstract

We introduce effective splitting methods for implementing optimization-based limiters to enforce the invariant domain in gas dynamics in high order accurate numerical schemes. The key ingredients include an easy and efficient explicit formulation of the projection onto the invariant domain set, and also proper applications of the classical Douglas-Rachford splitting and its more recent extension Davis-Yin splitting. Such an optimization-based approach can be applied to many numerical schemes to construct high order accurate, globally conservative, and invariant-domain-preserving schemes for compressible flow equations. As a demonstration, we apply it to high order discontinuous Galerkin schemes and test it on demanding benchmarks to validate the robustness and performance of both $\ell^1$-norm minimization limiter and $\ell^2$-norm minimization limiter.