Fractional Step Method for Singularly Perturbed 2D Delay Parabolic Convection Diffusion Problems on Shishkin Mesh

Abstract

In this article, we are interested to approximate the solution of a singularly perturbed 2D delay parabolic convection–diffusion initial-boundary-value problem. To discretize the continuous problem in the temporal direction, we use a fractional step method which results a set of two 1D problems. Next, we apply classical finite difference scheme on a special mesh to discretize those 1D problems in the spatial directions. Fractional step method for the time variable permits the computational cost reduction and the special mesh is used to capture the boundary layers. We derive the truncation errors for the scheme to obtain the error estimates, which shows that the scheme is uniformly convergent of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are presented to support the theoretical results.