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

Year 2019, Volume: 11 Issue: 4, 455 - 467, 05.12.2019

Tesfaye Bullo Gemechis Duressa , Guy Degla

https://doi.org/10.24107/ijeas.644160

Cited By: 5

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.

Keywords

Singularly perturbation parabolic problems, two parameters, fitted operator, accurate solution

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