Generic ADI Schemes with Time Order Constraints for Multidimensional Parabolic Equations in Finance

Abstract

In this paper, we provide a canonical polynomial formulation for the ADI schemes to solve multidimensional PDEs. We focus our analysis on the linear stability criteria applied to this formulation.We emphasize the importance of the time order: for any finite time order we characterize the set of specific constraints that must be verified in previously defined formulation. This allows therefore to construct a family of schemes for each time order. We discuss these resulting families and relate them to schemes typically found in the literature.We also investigate the application of the Richardson extrapolation to this family of schemes for achieving higher time order. This method is a sequence acceleration one.Based on this generic approach, we are moreover able to apply generic procedures to define an efficient scheme. For a fixed time order, we investigate the "fastest" (based on a simple computation cost model) and the "most accurate" (in the L1, L2 and L infinity norms) scheme. We expose the most interesting new schemes.Numerical illustrations are provided for various models and various classes of payoffs applied to different time orders.