An improved time accurate numerical estimation for singularly perturbed semilinear parabolic differential equations with small space shifts and a large time lag

Abstract

The objective of this work is to provide a robust numerical scheme for solving a singularly perturbed semilinear parabolic differential–difference equation having both time delay along with spatial delay, and shift parameters. The small delay and shift arguments in spatial direction are treated with Taylor’s approximation. The technique of quasilinearization is used to treat the semilinearity and the temporal direction is handled with the θ-method. To handle the layer behavior caused by the presence of perturbation parameter, the problem is solved using the upwind scheme on two-layer resolving meshes in the spatial direction providing a first-order accurate result for 0.5≤θ≤1. For θ=0.5, we have the second-order accurate Crank–Nicolson scheme in time. In order to have a higher-order accuracy, the Richardson extrapolation technique is applied in the spatial direction, which in turn provides a global second-order accurate result. Thomas algorithm is used to solve the tridiagonal system of equations formed in the fully discrete scheme. The numerical approach presented is shown to be parameter uniform and convergent for 0.5≤θ≤1. Some numerical tests are presented to show the efficacy of the proposed scheme.