Crank–Nicolson Method for Singularly Perturbed Unsteady Parabolic Problem With Multiple Boundary Turning Points

Abstract

In this paper, a numerical scheme for a time-dependent singularly perturbed parabolic convection–diffusion problem with boundary turning points is presented. The problem exhibits a left boundary layer in the spatial domain. We use the Crank–Nicolson method for temporal discretization and a nonstandard finite difference approach for spatial discretization on uniform meshes. Through rigorous error analysis, it has been shown that the scheme is stable and parameter-uniform convergence with a second-order accuracy in time and a first-order accuracy in space. Three model examples are provided to show the applicability of the scheme. It is shown that the numerical results are in agreement with the theoretical findings.