An invariant-region-preserving limiter for DG schemes to isentropic Euler equations

TL;DR

This paper develops an explicit invariant-region-preserving limiter for high-order DG schemes applied to the isentropic Euler equations, ensuring solutions stay within physical bounds without sacrificing accuracy.

Contribution

It introduces a novel IRP limiter compatible with high-order DG schemes for the p-system, maintaining invariant regions and accuracy.

Findings

The IRP limiter preserves the invariant region for smooth solutions.

The limiter does not compromise the scheme's order of accuracy.

Numerical tests confirm the effectiveness of the IRP DG schemes.

Abstract

In this paper, we introduce an invariant-region-preserving (IRP) limiter for the p-system and the corresponding viscous p-system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of the invariant region. 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 as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For any high order discontinuous Galerkin (DG) scheme to the p-system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscous p-system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes.