2015/08/18 by Maarten Wyns, Wyns, Maarten
Economics, Econometrics and Finance · Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1508.04296
openalex publication_date 2015/08/18 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
In this paper we consider the Modified Craig-Sneyd (MCS) scheme which forms a\nprominent time stepping method of the Alternating Direction Implicit type for\nmultidimensional time-dependent convection-diffusion equations with mixed\nspatial derivative terms. When the initial function is nonsmooth, which is\noften the case for example in financial mathematics, application of the MCS\nscheme can lead to spurious erratic behaviour of the numerical approximations.\nWe prove that this undesirable feature can be resolved by replacing the very\nfirst MCS timesteps by several (sub)steps of the implicit Euler scheme. This\ntechnique is often called Rannacher time stepping. We derive a useful\nconvergence bound for the MCS scheme combined with Rannacher time stepping when\nit is applied to a model two-dimensional convection-diffusion equation with\nmixed-derivative term and with Dirac-delta initial data. Ample numerical\nexperiments are provided that show the sharpness of our obtained error bound.\n