Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations

TL;DR

This paper introduces an explicit invariant-region-preserving limiter for high-order discontinuous Galerkin schemes solving multi-dimensional hyperbolic conservation laws, ensuring solutions stay within physical bounds without sacrificing accuracy.

Contribution

It develops a new IRP limiter compatible with high-order DG methods for multi-dimensional systems, with proven accuracy preservation and applicability to Euler equations.

Findings

IRP limiter maintains solutions within invariant regions.

High-order IRP DG schemes are validated for Euler equations.

Limiter does not compromise approximation accuracy.

Abstract

An invariant-region-preserving (IRP) limiter for multi-dimensional hyperbolic conservation law systems is introduced, as long as the system admits a global invariant region which is a convex set in the phase space. It is shown that the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average is away from the boundary of the convex set. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes. For arbitrarily high order discontinuous Galerkin (DG) schemes to hyperbolic conservation law systems, sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one-dimensional system shares the same invariant region as the full multi-dimensional hyperbolic system {does}. The general results are then applied to both one and two dimensional compressible Euler equations so to obtain high order IRP DG schemes. Numerical experiments are provided to validate the proven properties of the IRP limiter and the performance of IRP DG schemes for compressible Euler equations.