Convergence of the Modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term

TL;DR

This paper proves a convergence theorem for the Modified Craig-Sneyd scheme applied to 2D convection-diffusion equations with mixed derivatives, showing error bounds independent of spatial mesh size, supported by numerical experiments.

Contribution

The paper provides the first convergence proof for the MCS scheme with error bounds independent of spatial discretization size in 2D convection-diffusion equations with mixed derivatives.

Findings

Error bounds are independent of spatial mesh width.

Numerical experiments confirm theoretical convergence.

Applicable to financial mathematics models.

Abstract

We consider the Modified Craig-Sneyd (MCS) scheme which forms a prominent time stepping method of the Alternating Direction Implicit type for multidimensional time-dependent convection-diffusion equations with mixed spatial derivative terms. Such equations arise often, notably, in the field of financial mathematics. In this paper a first convergence theorem for the MCS scheme is proved where the obtained bound on the global temporal discretization errors has the essential property that it is independent of the (arbitrarily small) spatial mesh width from the semidiscretization. The obtained theorem is directly pertinent to two-dimensional convection-diffusion equations with mixed derivative term. Numerical experiments are provided that illustrate our result.