Parametrized Maximum Principle Preserving Limiter for Finite Difference WENO Schemes Solving Convection-Dominated Diffusion Equations

Abstract

In this paper, we generalize the parametrized maximum principle preserving flux limiting technique, designed for the high order WENO method of solving scalar hyperbolic conservation law problems, to convection-dominated diffusion computation. We modify the high order numerical flux by limiting it toward a lower order monotone flux to develop a maximum principle preserving high order finite difference Runge--Kutta WENO method for the convection-dominated diffusion problems. The proposed method has several advantages. First, the flux limiting technique can be conveniently applied to arbitrarily high order Runge--Kutta WENO schemes. It requires only conservative discretization of both convection and diffusion terms, which is a natural and standard procedure. Second, the flux limiting technique does not demand much time step restriction. For general nonlinear convection-dominated diffusion problems, formal analysis shows that a third order finite difference scheme with parametrized maximum principle preserving flux limiters preserves third order accuracy without extra CFL constraints when the lower order monotone flux is made of local Lax--Friedrich flux for a convection term and a second order central difference for diffusion term. For schemes with higher than third order accuracy, extensive numerical tests are performed to demonstrate the effectiveness of the proposed method. The generalized flux limiters are applied to high order finite difference WENO computation of the incompressible Navier--Stokes equations in vorticity stream-function formulation and several other problems. Numerical results are given for both one- and two-dimensional problems.