Maximum-Principle-Preserving and Positivity-Preserving Central WENO Schemes on Overlapping Meshes

Abstract

In this paper, we propose a class of maximum-principle-preserving central WENO schemes for scalar conservation laws, and positivity-preserving central WENO schemes for compressible Euler equations. Formulated in a finite volume framework on overlapping meshes, the central schemes require neither flux splitting nor numerical fluxes that are often exact or approximate Riemann solvers. A new fifth-order WENO reconstruction is applied for the spatial discretization, and the linear weights of such reconstruction can be any positive number as long as their sum equals one, which leads to much simpler implementation. The sufficient conditions are provided for the cell average values to preserve maximum principle or positivity property with Euler forward time discretization. The method can be generalized to high-order strong stability preserving Runge-Kutta method without technical difficulties. Extensive numerical examples are presented to illustrate the accuracy and performance of the proposed methods.