Discretization of Anisotropic Convection-diffusion Equations, Convective M-matrices and their Iterative Solution

Abstract

We derive the constant-j box method discretization for the convection-diffusion equation, ∇j=f, with j=−α∇u+βu . In two dimensions, α is a 2 × 2 symmetric, positive definite tensor field and β is a two-dimensional vector field. This derivation generalizes the well-known Scharfetter-Gummel discretization of the continuity equations in semiconductor device simulation. We define the anisotropic Delaunay condition and show that under this condition and appropriate evaluations of α and β, the stiffness matrix, M, of the discretization is a convective M-matrix. We then examine classical iterative splittings of M and show that convection (even convection dominance) does not degrade the rate of convergence of such iterations relative to the purely diffusive (β=0) problem under certain conditions.