Non-standard finite-difference scheme for singularly perturbed parabolic convection-diffusion problem with boundary turning points

Document Type : Research Article

Authors

Department of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia

Abstract

This paper presents a numerical method for solving singularly perturbed parabolic convection-diffusion problems with boundary turning points. As the perturbation parameter $\varepsilon$ approaches zero, the solution shows rapid changes on the left side of the spatial domain, forming a small boundary layer. The classical finite difference methods on uniform meshes fail to capture these oscillations without using a large number of mesh points. To solve this, we use the implicit Euler method for time discretization and a non-standard finite difference method in space. The method satisfies stability, the discrete minimum principle, and $\varepsilon$-uniform convergence. Error estimates show that the proposed method is first-order convergence in time and space. The order of convergence is improved by applying the Richardson extrapolation method. Two model examples are provided to show the scheme's applicability. It demonstrates that the numerical results are in agreement with the theoretical findings.

Keywords

Main Subjects


[1] M.A. Ayele, A.A. Tiruneh, G.A. Derese, Uniformly convergent scheme for singularly perturbed
space delay parabolic differential equation with discontinuous convection coefficient and source
term, J. Math. 2022 (2022) 1874741.
[2] R.K. Dunne, E. ORiordan, G.I. Shishkin, A Fitted mesh method for a class of singularly perturbed
parabolic problems with a boundary turning point, Comput. Methods Appl. Math. 3 (2003) 361–
372.
[3] T.C. Hanks, Model relating heat-flow values near, and vertical velocities of mass transport beneath,
oceanic Rises, J. Geophys.Res. 76 (1971) 537–544.
[4] Z.I. Hassen, G.F. Duressa, Nonstandard hybrid numerical scheme for singularly perturbed
parabolic differential equations with large delay, Partial Differ. Equ. Appl. Math. 10 (2024)
100722.
[5] A. Kaushik, N. Sharma, An adaptive difference scheme for parabolic delay differential equation
with discontinuous coefficients and interior layers, J. Differ. Equ. Appl. 26 (2020) 1450–1470.
[6] R.B. Kellogg, A. Tsan, Analysis of some difference approximations for a singular perturbation
problem without turning points, Math. Comput. 32 (1976) 1025–1039.
[7] P. Kumari, D. Kumar, H. Ramos, Parameter independent scheme for singularly perturbed problems
including a boundary turning point of multiplicity greater than or equal to one, J. Appl. Anal.
Comput. 13 (2023) 1304–1320.
[8] S. Kumar, Sumit, J. Vigo-Aguiar, A parameter-uniform grid equidistribution method for singularly
perturbed degenerate parabolic convection–diffusion problem, J. Comput. Appl. Math. 404 (2022)
113273.
[9] H.G. Kumie , A.A. Tiruneh G.A. Derese, Crank–Nicolson method for solving time-fractional sin-
gularly perturbed delay partial differential equations, Res. Math. 11 (2024) 2293373.
[10] S. Ku Sahoo, V. Gupta, Second-order parameter-uniform finite difference scheme for singularly
perturbed parabolic problem with a boundary turning point, J. Differ. Equ. Appl. 27 (2021) 223–
240.
[11] A. Majumdar, S. Natesan, An ε-uniform hybrid numerical scheme for a singularly perturbed de-
generate parabolic convection–diffusion problem, Int. J. Comput. Math. 96 (2019) 1313–1334.
[12] A. Majumdar, S. Natesan, Second-order uniformly convergent Richardson extrapolation method for
singularly perturbed degenerate parabolic PDEs, Int. J. Appl. Comput. Math. 3 (2017) 31–53.
[13] W.G. Melesse, A.A. Tiruneh, G.A. Derese, Fitted mesh method for singularly perturbed delay dif-
ferential turning point problems exhibiting twin boundary layers, J. Appl. Math. Inform. 38 (2020)
113–132.
[14] R.E. Mickens, Advances in the Applications of Nonstandard Finite Difference Scheme, World Sci-
entific, 2005.
[15] J.J. H. Miller, E.O. Riordan, On piece-wise uniform meshes for upwind- and central-difference
operators for solving singularly perturbed problems, IMA J. Numer. Anal. 15 (1995) 89–99.
[16] P. Rai, S. Yadav, Robust numerical schemes for singularly perturbed delay parabolic convection-
diffusion problems with degenerate coefficient, Int. J. Comput. Math. 98 (2021) 195–221.
[17] S.J. Polak, C. Den Heijer, W.H. A. Schilders, P. Markowich, Semiconductor device modelling from
the numerical point of view, Int. J. Numer. Methods Eng. 24 (1987) 763–838.
[18] C. Sangeetha, A. Awasthi, A robust numerical method for solving time dependent singularly per-
turbed two parameter problem using non-standard finite difference method, Asian-European J.
Math. (2024) 2450084.
[19] K. Sharma, P. Rai, K.C. Patidar, A review on singularly perturbed differential equations with turning
points and interior layers, Appl. Math. Comput. 219 (2013) 10575–10609.
[20] G.I. Shishkin, Grid approximations to singularly perturbed parabolic equations with turning
points, Differ. Equ. 37 (2001) 1037–1050.
[21] S.K. Tesfaye, G.F. Duressa, M.M. Woldaregay, T.G. Dinka, A uniformly convergent numerical
scheme for singularly perturbed parabolic turning point problem, J. Math. Model. 12 (2024) 501–
516.
[22] A.A. Tiruneh, G.A. Derese, D.M. Tefera, A nonstandard fitted operator method for singularly per-
turbed parabolic reaction-diffusion problems with a large time delay, Int. J. Math. Math. Sci. 2022
(2022) 5625049.
[23] R. Vulanovic, P.A. Farrell, Continuous and numerical analysis of a multiple boundary turning point
problem, SIAM J. Numer. Anal. 30 (1993) 1400–1418.
[24] S. Yadav, P. Rai, K. Sharma, A higher order uniformly convergent method for singularly perturbed
parabolic turning point problems, Numer. Methods Partial Differ. Equ. 36 (2020) 342–368.
[25] A.R. Yaghoubi, H.S. Najafi, Non-standard finite difference schemes for investigating stability of a
mathematical model of virus therapy for cancer, Appl. Appl. Math. Int. J. 14 (2019) 11.
[26] A.R. Yaghoubi, H.S. Najafi, A fully implicit non-standard finite difference scheme for one dimen-
sional Burgers’ equation, J. Appl. Res. Ind. Eng. 7 (2020) 301–312.