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

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.