Adaptive mesh methods for hyperbolic conservation laws with bound-preserving flux limiters

TL;DR

This paper introduces bound-preserving finite-volume schemes for hyperbolic conservation laws on adaptive moving meshes, ensuring high accuracy and robustness while maintaining physical bounds through efficient flux limiters.

Contribution

It develops a novel bound-preserving flux limiter approach compatible with adaptive moving meshes, extending to nonlinear systems like Euler equations and two-medium flow models.

Findings

Schemes achieve high resolution and robustness.

Bound-preserving property is maintained under mild CFL conditions.

Method extends to complex nonlinear systems.

Abstract

In this paper, we develop bound-preserving (BP) finite-volume schemes for hyperbolic conservation laws on adaptive moving meshes. For scalar conservative laws, we rewrite the conventional high-order discretization as a convex combination of first-order counterparts on each sub-cell, which is mathematically equivalent to introducing a bound-preserving flux limiter. Such a limiter is inexpensive to evaluate, with a feature that the corresponding BP CFL conditions depend solely on the first-order sub-cell schemes. A mild CFL restriction is derived under which high-order spatial accuracy is retained. The proposed BP schemes are extend to two nonlinear systems, namely, the Euler equations and the five-equation transport model of two-medium flows. Numerical results demonstrate that the present schemes possess high resolution and strong robustness properties.