High Order Uniformly Convergent Fractional Step RK Methods and HODIE Finite Difference Schemes for 2D Evolutionary Convection-Diffusion Problems

Abstract

In this work we present a numerical method to solve linear time dependent two dimensional singularly perturbed problems of convection-diffusion type with dominating convection term; this class of problems is characterized by the presence of a regular boundary layer in the output boundary of the spatial domain. The method combines the alternating direction technique, based on an A-stable third order RK method, with a third order HODIE finite difference scheme of classical type, i.e., exact only on polynomial functions, constructed on a special spatial mesh of Shishkin type. We show that, under appropriate restrictions between the discretization parameters, the method is uniformly convergent with respect to the diffusion parameter, having order three (except by a logarithmic factor) in the maximum norm. The method provides the computational advantages of the splitting technique and also the efficiency provided by high order methods, achieving good approximations of the solution in the whole domain, including the boundary layer region. We show some numerical results validating in practice the good properties of the method.