Uniformly Convergent Numerical Method for Two-parametric ‎Singularly Perturbed Parabolic Convection-diffusion Problems

Abstract

This paper deals with the numerical treatment of two-parametric singularly perturbed parabolic convection-diffusion problems. The scheme is developed through the Crank-Nicholson discretization method in the temporal dimension followed by fitting the B-spline collocation method in the spatial direction. The effect of the perturbation parameters on the solution profile of the problem is controlled by fitting a parameter. As a result, it has been observed that the method is a parameter-uniform convergent and its order of convergence is two. This is shown by the boundedness of the solution, its derivatives, and error estimation. The effectiveness of the proposed method is demonstrated by model numerical examples, and more accurate solutions are obtained as compared to previous findings available in the literature.

Keywords

• singularly perturbed
• parabolic convection-diffusion
• b-spline collocation
• parameter-uniform