A Numerical Approach for Diffusion-Dominant Two-Parameter Singularly Perturbed Delay Parabolic Differential Equations

Abstract

A numerical scheme is developed to solve a large time delay two-parameter singularly perturbed one-dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece-wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second-order convergent in the spatial direction and second-order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.