## Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws.(English)Zbl 1286.65102

Summary: We 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 are 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 the strict maximum principle with the mild Courant-Friedrichs-Lewy constraint. Two-dimensional numerical evidence is given to demonstrate the capability of the proposed approach.

### MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 35B50 Maximum principles in context of PDEs
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