  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.
