A uniformly convergent alternating direction HODIE finite difference scheme for 2D time-dependent convection-diffusion problems

摘要

In this work, we design and analyze a finite difference scheme used to solve a class of 2D time-dependent convection-diffusion problems, for which we suppose that the convection term is positive in both spatial directions. We use the Peaceman and Rachford method to discretize in time such problems and high-order differences via an identity expansion finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space. We prove that the method is uniformly convergent with respect to the diffusion parameter, reaching almost order two in space. A brief discussion concerning the theoretical and practical orders of convergence in time is included, pointing out possible theoretical advances in the future. We present some numerical examples illustrating such a behaviour; they indicate that the numerical method is also suitable in a wider set of singularly perturbed problems than the ones defined by the theoretical restrictions.