A second order fractional step hybrid numerical algorithm for time delayed singularly perturbed 2D convection-diffusion problems

Abstract

The objective of this work is to provide an efficient and higher-order numerical scheme for the solution of a singularly perturbed 2D time-delayed parabolic convection-diffusion problem. Each time step is split into two partial time steps by using the Peaceman-Rachford splitting algorithm. The splitting is done over the idea of the Crank-Nicholson scheme. The hybrid scheme is a combination of the central difference scheme in the layer region and the mid-point upwind scheme in the outer region defined on a Shishkin mesh for the discretization in the spatial direction. This makes the order of convergence to two (up to a logarithmic factor) in space. Again, the presence of the logarithmic effect is rectified by the use of the Bakhvalov-Shishkin mesh. The proposed method is proved to be parameter uniform and also it attains second-order spatial accuracy in the discrete supremum norm. Numerical tests validate the efficacy of the proposed scheme.