Convergence of the Hundsdorfer–Verwer scheme for two-dimensional convection-diffusion equations with mixed derivative term

Abstract

Alternating Direction Implicit (ADI) schemes are popular in the numerical solution of multidimensional time-dependent partial differential equations (PDEs) arising in various contemporary application fields such as financial mathematics. The Hundsdorfer–Verwer (HV) scheme is an often used ADI scheme. A structural analysis of its fundamental properties, notably convergence, is of main interest. Up to now, however, a convergence result is only known in the literature relevant to the case of one-dimensional PDEs. In this paper we prove that, under natural stability and smoothness conditions, the HV scheme has a temporal order of convergence equal to two, uniformly in the spatial mesh width, whenever it is applied to two-dimensional convection-diffusion equations with mixed derivative term.