A uniformly convergent quadratic -spline based scheme for singularly perturbed degenerate parabolic problems

Abstract

In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme.