Higher Order Fitted Operator Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems

Abstract

In this paper, we consider singularly perturbed parabolic convection-diffusion initial boundary value problems with two small positive parameters to construct higher order fitted operator finite difference method. At the beginning, we discretize the solution domain in time direction to approximate the derivative with respect to time and considering average levels for other terms that yields two point boundary value problems which covers two time level. Then, full discretization of the solution domain followed by the derivatives in two point boundary value problem are replaced by central finite difference approximation, introducing and determining the value of fitting parameter ended at system of equations that can be solved by tri-diagonal solver. To improve accuracy of the solution with corresponding higher orders of convergence, we applying Richardson extrapolation method that accelerates second order to fourth order convergent. Stability and consistency of the proposed method have been established very well to assure the convergence of the method. Finally, validate by considering test examples and then produce numerical results to care the theoretical results and to establish its effectiveness. Generally, the formulated method is stable, consistent and gives more accurate numerical solution than some methods existing in the literature for solving singularly perturbed parabolic convection- diffusion initial boundary value problems with two small positive parameters.