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

Year 2019, , 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

References

• References[1]. Das P. and Mehrmann V., Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters, BIT Numer Math, DOI 10.1007/s10543-015-0559-8, 2015.
• [2]. Gupta V., Kadalbajoo M. K. and Dubey R. K., A parameter uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2018.1432856, 2018.
• [3]. Jha A. and Kadalbajoo M. K., A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems, International Journal of Computer Mathematics, Taylor and Frances group, 92 1204–1221, 2015.
• [4]. Kadalbajoo M. K. and Yadaw A.S., Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems, Int. J. Comput. Methods, 9, 1250047-1 - 1250047-16, 2012.
• [5]. Kumar V. and Srinivasan B., A novel adaptive mesh strategy for singularly perturbed parabolic convection diffusion problems, Differ Equ Dyn Syst, DOI 10.1007/s12591-017-0394-2, 2017.
• [6]. Miller H. J.J, O’Riordan E. and Shishkin I. G., Fitted numerical methods for singular perturbation problems, Error estimate in the maximum norm for linear problems in one and two dimensions, World Scientific, 1996.
• [7]. Morton K. W., Numerical solution of convection-diffusion problems, CRC Press, Taylor and Francis group, 1996.
• [8]. Munyakazi J. B., A Robust Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems, An International Journal of Applied Mathematics & Information Sciences, 9, 2877-2883, 2015.
• [9]. Roos G. H., Stynes M.and Tobiska L., Robust numerical methods for singularly perturbed differential equations, Convection-Diffusion-Reaction and Flow Problems, Springer Series in Computational Mathematics, 2008.
• [10]. Smith G. D., Numerical solution of partial differential equations, Finite difference methods, Third edition, Oxford University Pres, 1985.
• [11]. Suayip Y. and S. Niyazi S., Numerical solutions of singularly perturbed one-dimensional parabolic convection–diffusion problems by the Bessel collocation method, Applied Mathematics and Computation, 220, 305–315, 2013.
• [12]. Zahra W. K., El-Azab M. S. and El Mhlawy A. M. (2014), Spline difference scheme for two-parameter singularly perturbed partial differential equations, J. Appl. Math. and Informatics, 32, 185 – 201, 2014.
• [13]. Zhilin L., Qiao Z. and Tang T. Numerical solution of differential equations, Introduction to finite difference and finite element methods, printed in the United Kingdom by Clays, 2018.