High Order Numerical Methods Preserving Invariant Domain for Hyperbolic and Related Systems

TL;DR

This paper surveys invariant domain preserving (IDP) numerical schemes for hyperbolic systems, emphasizing recent high-order methods that ensure physically meaningful solutions and numerical stability through innovative limiting techniques.

Contribution

It provides a systematic review of IDP schemes, introduces new perspectives on high-order methods, and compares polynomial limiters and flux limiting approaches for complex hyperbolic systems.

Findings

High-order IDP schemes effectively preserve physical invariants.

Polynomial limiters enforce strong IDP properties in high-order schemes.

Flux limiting approaches adapt well to various discretizations.

Abstract

Admissible states in hyperbolic systems and related equations often form a convex invariant domain. Numerical violations of this domain can lead to loss of hyperbolicity, resulting in illposedness and severe numerical instabilities. It is therefore crucial for numerical schemes to preserve the invariant domain to ensure both physically meaningful solutions and robust computations. For complex systems, constructing invariant-domain-preserving (IDP) schemes is highly nontrivial and particularly challenging for high-order accurate methods. This paper presents a comprehensive survey of IDP schemes for hyperbolic and related systems, with a focus on the most popular approaches for constructing provable IDP schemes. We first give a systematic review of the fundamental approaches for establishing the IDP property in first-order accurate schemes, covering finite difference, finite volume, finite element, and residual distribution methods. Then we focus on two widely used and actively developed classes of high order IDP schemes as well as their recent developments, most of which have emerged in the past decade. The first class of methods seeks an intrinsic weak IDP property in high-order schemes and then designs polynomial limiters to enforce a strong IDP property at the points of interest. This generic approach applies to high-order finite volume and discontinuousGalerkin schemes. The second class is based on the flux limiting approaches, which originated from the flux-corrected transport method and can be adapted to a broader range of spatial discretizations, including finite difference and continuous finite element methods. In this survey, we elucidate the main ideas in the construction of IDP schemes, provide some new perspectives and insights, with extensive examples, and numerical experiments in gas dynamics and magnetohydrodynamics.