A second order numerical method for singularly perturbed delay parabolic partial differential equation

Abstract

Purpose The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Design/methodology/approach For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme. Findings The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time. Originality/value A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.