Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems

TL;DR

This paper presents a discretization-independent, invariant domain preserving scheme for nonlinear hyperbolic systems, introducing convex limiting to ensure high-order accuracy while maintaining essential solution bounds.

Contribution

It introduces a novel approximation technique that preserves invariant domains for hyperbolic systems and develops convex limiting to correct higher-order methods.

Findings

The scheme is discretization-independent under certain symmetry conditions.

Convex limiting effectively enforces invariant domain properties.

High-order methods can be corrected to satisfy invariants without losing accuracy.

Abstract

We introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving. The method is discretization-independent provided elementary symmetry and skew-symmetry properties are satisfied by the scheme. The method is formally first-order accurate in space. A series of higher-order methods is also introduced. When these methods violate the invariant domain properties, they are corrected by a limiting technique that we call convex limiting. After limiting, the resulting methods satisfy all the invariant domain properties that are imposed by the user (see Theorem~7.24). A key novelty is that the bounds that are enforced on the solution at each time step are necessarily satisfied by the low-order approximation.