Grid-Free Redistribution Methods for Axisymmetric and Anisotropic Diffusion

Abstract

The Vorticity Redistribution Method developed in this report is a numerical method to handle partial differential equations arising out of modeling diffusion processes. The redistribution method is essentially based on the evolution of integral invariants of a partial differential equation. For certain class of physical processes, such an evolution inherently allows the method to accurately compute the numerical solutions better than the traditional methods, for example, a finite difference method. Although the redistribution method was developed for handling diffusion processes in aerodynamic flows, it is equally applicable to some problems in finance. Indeed, recently, a one-dimensional redistribution method is used to compute the forward Kolmogorov equation for transition density to value the european option under stochastic interest The actual development and application of the redistribution method to the option pricing problem is in the author's report titled Forward density approach to european option under stochastic interest rate. Since the initial condition for the Kolmogorov equation is a delta function, the redistribution method is perfectly suited to handle the delta function without the need to regularize it as is usually done in a finite difference computation. The redistribution and finite difference methods are both used to compute the Kolmogorov equation. The results show that the redistribution method better handles the common difficulties encountered in a finite difference method, such as the errors due to low mesh resolution and far-field boundary truncation.