Discretization of Anisotropic Convection-diffusionEquations, Convective M-matrices and their IterativeSolution

[摘要] We derive the constant-jbox method discretization for the convection-diffusionequation,∇j=f, withj=−α∇u+βu. In two dimensions,αis a 2 × 2 symmetric, positivedefinite tensor field andβis a two-dimensional vector field. This derivation generalizesthe well-known Scharfetter-Gummel discretization of the continuity equations insemiconductor device simulation. We define the anisotropic Delaunay condition andshow that under this condition and appropriate evaluations ofαandβ, the stiffnessmatrix,M, of the discretization is a convectiveM-matrix. We then examine classicaliterative splittings ofMand show that convection (even convection dominance) doesnot degrade the rate of convergence of such iterations relative to the purely diffusive (β=0) problem under certain conditions.

[学科分类] 电子、光学、磁材料