On the Discrete Maximum Principle for Algebraic Flux Correction Schemes with Limiters of Upwind Type

Abstract

Algebraic flux correction (AFC) schemes are applied to the numerical solution of scalar steady-state convection-diffusion-reaction equations. A general result on the discrete maximum principle (DMP) is established under a weak assumption on the limiters and used for proving the DMP for a particular limiter of upwind type under an assumption that may hold also on non-Delaunay meshes. Moreover, a simple modification of this limiter is proposed that guarantees the validity of the DMP on arbitrary simplicial meshes. Furthermore, it is shown that AFC schemes do not provide sharp approximations of boundary layers if meshes do not respect the convection direction in an appropriate way.