Collocation method using artificial viscosity for time dependent singularly perturbed differential–difference equations

Abstract

A parameter uniform numerical method is presented for solving singularly perturbed time-dependent differential–difference equations with small shifts. To approximate the terms with the shifts, Taylor’s series expansion is used. The resulting singularly perturbed parabolic partial differential equation is solved using an implicit Euler method in temporal direction and cubic B-spline collocation method for the resulting system of ordinary differential equations in spatial direction, and an artificial viscosity is introduced in the scheme using the theory of singular perturbations. The proposed method is shown to be accurate of order OΔt+h2 by preserving ɛ-uniform convergence, where h and Δt denote spatial and temporal step sizes, respectively. Several test examples are solved to demonstrate the effectiveness of the proposed method. The computed numerical results show that the proposed method provides more accurate results than some methods exist in the literature and suitable for solving such problems with little computational effort.