Diagonal Dominance and Positive Definiteness of Upwind Approximations for Advection Diffusion Problems

Abstract

We examine whether three different upwind schemes for the positive definite advection diffusion problem can yield indefinite coefficient matrices. In particular, in line with the increasing use of adaptive meshes, we are concerned with discretisations on arbitrary grids but only in one dimension. We show that indefinite coefficient matrices can arise from certain approaches: this can present difficulties with efficient iterative solution techniques which might be required for corresponding approximations in higher dimensions.

We dedicate this paper to Professor Ron Mitchell who has been an inspiration to many generations of numerical analysts. In addition his generous spirit has infused our discipline with a sense of positive endeavour.