A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

Abstract

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091---3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752---2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge---Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.