Parametrized Maximum Principle Preserving Flux Limiters for High Order Schemes Solving Multi-Dimensional Scalar Hyperbolic Conservation Laws

Abstract

In this paper, we will extend the strict maximum principle preserving flux limiting technique developed for one dimensional scalar hyperbolic conservation laws to the two-dimensional scalar problems. The parametrized flux limiters and their determination from decoupling maximum principle preserving constraint is presented in a compact way for two-dimensional problems. With the compact fashion that the decoupling is carried out, the technique can be easily applied to high order finite difference and finite volume schemes for multi-dimensional scalar hyperbolic problems. For the two-dimensional problem, the successively defined flux limiters are developed for the multi-stage total-variation-diminishing Runge---Kutta time-discretization to improve the efficiency of computation. The high order schemes with successive flux limiters provide high order approximation and maintain strict maximum principle with mild Courant-Friedrichs-Lewy constraint. Two dimensional numerical evidence is given to demonstrate the capability of the proposed approach.