Second-order invariant domain preserving approximation of the Euler equations using convex limiting

TL;DR

This paper introduces a second-order numerical method for the Euler equations that preserves key physical invariants like density positivity and entropy, combining a first-order invariant domain method with a high-order approach through convex limiting.

Contribution

A novel second-order invariant domain preserving method for Euler equations that integrates a guaranteed maximum speed approach with high-order schemes using convex limiting.

Findings

Achieves second-order accuracy in maximum norm.

Preserves positivity of density and internal energy.

Convex limiting is adaptable to other methods and systems.

Abstract

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.