Fitted Operator Average Finite Difference Method for Singularly Perturbed Parabolic Convection- Diffusion Problem

Abstract

In this paper, we study a fitted operator average finite difference method for solving singularly perturbed parabolic convection-diffusion problems with boundary layer at right side. After discretizing the solution domain uniformly, the differential equation is replaced by average finite difference approximation which gives system of algebraic equation at each time levels. The stability and consistency of the method established very well to guarantee the convergence of the method. Furthermore, some numerical results are given to support our theoretical results and to validate the betterment of using fitted operator methods

Keywords

• Fitted operator,singular perturbation,parabolic problems

References

1. [1] Gowrisankar S., Srinivasan N., Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids, Computer Physics Communications, 185, 2008-2019, 2014
2. [2] Munyakazi J. B., A Robust Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems, An International Journal of Applied Mathematics & Information Sciences, Vol. 9(6), 2877-2883, 2015
3. [3] 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
4. [4] 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
5. [5] Rai P. and. Sharma K. K., Singularly perturbed parabolic differential equations with turning point and retarded arguments, IAENG International Journal of Applied Mathematics, 45:4, IJAM_45_4_20, 2015
6. [6] Mohanty R. K., Dahiya V., Khosla N., Spline in Compression Methods for Singularly Perturbed 1D Parabolic Equations with Singular Coefficients, Open Journal of Discrete Mathematics, 2, 70-77, 2012
7. [7] Roos G. H., Stynes M.and Tobiska L., Robust numerical methods for singularly perturbed differential equations, Convection-diffusion-reaction and flow problems, Springer-Verlag Berlin Heidelberg, Second Edition, 2008
8. [8] Suayip Y. S. and Sahin N., Numerical solutions of singularly perturbed one-dimensional parabolic convection–diffusion problems by the Bessel collocation method, Applied Mathematics and Computation 220, 305–315, 2013 [9] Vivek K. 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 [10]. Yanping C. and Li-Bin L., An adaptive grid method for singularly perturbed time – dependent convection diffusion problems, Commun. Comput. Phys, 20, 1340-1358, 2016.