An invariant-region-preserving limiter for DG schemes to isentropic Euler equations
This work provides a practical, provably accurate limiter for preserving invariant regions in high-order DG schemes for hyperbolic conservation laws, addressing a known bottleneck in computational fluid dynamics.
The authors introduce an invariant-region-preserving (IRP) limiter for DG schemes applied to the p-system and viscous p-system, proving that it maintains high-order accuracy for smooth solutions and providing explicit, easy-to-implement algorithms. Numerical experiments validate the limiter's properties.
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.