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

We derive the constant-j box method discretization for the convection-diiusion equation, rj = f, with j = ?ru + u. In two dimensions, is a 2 2 symmetric, positive deenite tensor eld and is a two-dimensional vector eld. This derivation generalizes the well-known Scharfetter-Gummel discretization of the continuity equations in semiconductor device simulation. We deene the anisotropic Delaunay condition and show that under this condition and appropriate evaluations of and , the stiiness 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 diiusive (= 0) problem under certain conditions.