A family of finite difference schemes for the convection-diffusion equation in two dimensions

Abstract

The construction of finite difference schemes in two dimensions is more ambiguous than in one dimension. This ambiguity arises because different combinations of local nodal values are equally able to model local behaviour with the same order of accuracy. In this paper we outline an evolutionary operator for the two dimensional convection-diffusion problem in an unbounded domain and use it as the source for obtaining a family of second order (Lax-Wendroff) schemes and third-order (Quickest) schemes not yet studied in the literature. Additionally we study in detail the stability of those second-order and third-order schemes, a crucial property for convergence of numerical schemes, using the von Neumann method.