A new stability result for the modified Craig–Sneyd scheme applied to two-dimensional convection–diffusion equations with mixed derivatives

Abstract

The Modified Craig–Sneyd scheme is an alternating direction implicit(ADI) type scheme that was introduced by In ’t Hout and Welfert (2009) [12] in order to numerically solve multidimensional convection–diffusion equations with mixed-derivative terms. It is one of the most prominent ADI schemes currently known for their efficiency in solving above type of problems. This paper deals with a useful stability result for the Modified Craig–Sneyd scheme when applied to two-dimensional convection–diffusion equations with mixed derivative term. The stability of the scheme is analyzed in the von Neumann framework, effectively taking into account the actual size of the mixed derivative term. This study is relevant to an observation of apparent discrepancy in a real world application of the scheme, i.e., in computational finance. The obtained results not only generalize some of the existing stability results, but also clearly justify this surprising observation theoretically.