Convergence analysis of the Modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with nonsmooth initial data

TL;DR

This paper analyzes the convergence of the Modified Craig-Sneyd scheme for 2D convection-diffusion equations with nonsmooth initial data, proposing Rannacher time stepping to improve stability and providing theoretical and numerical validation.

Contribution

It introduces a convergence analysis for the MCS scheme with Rannacher time stepping on nonsmooth initial data, offering new error bounds and practical insights.

Findings

Rannacher time stepping improves stability for nonsmooth initial data.

Derived sharp convergence bounds for the combined scheme.

Numerical experiments confirm theoretical error estimates.

Abstract

In this paper 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. When the initial function is nonsmooth, which is often the case for example in financial mathematics, application of the MCS scheme can lead to spurious erratic behaviour of the numerical approximations. We prove that this undesirable feature can be resolved by replacing the very first MCS timesteps by several (sub)steps of the implicit Euler scheme. This technique is often called Rannacher time stepping. We derive a useful convergence bound for the MCS scheme combined with Rannacher time stepping when it is applied to a model two-dimensional convection-diffusion equation with mixed-derivative term and with Dirac-delta initial data. Ample numerical experiments are provided that show the sharpness of our obtained error bound.