Convergence analysis of the modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with nonsmooth initial data. (English) Zbl 1433.65169
Summary: In this article 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.
MSC:
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65L04 | Numerical methods for stiff equations |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
35K57 | Reaction-diffusion equations |
91G60 | Numerical methods (including Monte Carlo methods) |