A fitted operator method of line scheme for solving two-parameter singularly perturbed parabolic convection-diffusion problems with time delay

This paper presents a parameter-uniform numerical scheme for the solution of two-parameter singularly perturbed parabolic convection-diffusion problems with a delay in time. The continuous problem is semi-discretized using the Crank-Nicolson finite difference method in the temporal direction. The resulting differential equation is then discretized on a uniform mesh using the fitted operator finite difference method of line scheme. The method is shown to be accurate in O(\left(\Delta t \right)^{۲}  + N^{-۲}) , where N is the number of mesh points in spatial discretization and \Delta t is the mesh length in temporal discretization. The parameter-uniform convergence of the method is shown by establishing the theoretical error bounds. Finally, the numerical results of the test problems validate the theoretical error bounds.

Keywords:

Singular perturbation , time-delayed parabolic convection-diffusion problems , two small parameters , the method of line , finite difference scheme , uniform convergence