New Stability Results of the Modified Craig-Sneyd Scheme in a Multidimensional Diffusion Equation with Mixed Derivative Terms

Abstract

The time-dependent multidimensional diffusion equations with mixed derivative terms have been widely used in mathematics. Due to the mixed derivative terms, it is difficult to solve this pattern of multidimensional diffusion equations. The Modified Craig-Sneyd scheme (MCS), which has been applied to value the power derivatives, is one of the most promising methods for solving such equations. Therefore, it is of great importance to study its stability. The unconditional stability problem of MCS for a multidimensional diffusion equation with mixed derivative terms is investigated by taking into account the sizes. A new sufficient condition, necessary condition, and equivalence condition on the parameter of the MCS scheme for unconditional stability are proposed in case of the two and three-dimensional diffusion equations.