An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ

Ram Shiromani; Carmelo Clavero; Ram Shiromani; Carmelo Clavero

doi:10.3934/math.20241688

AIMS Mathematics

2024, Volume 9, Issue 12: 35570-35598. doi: 10.3934/math.20241688

Previous Article Next Article

Research article Special Issues

An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ

Ram Shiromani ^{1
,
,},
Carmelo Clavero ²

1.
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Uttar Pradesh, India
2.
Department of Applied Mathematics, IUMA, University of Zaragoza, Zaragoza, Spain

Received: 25 September 2024 Revised: 11 December 2024 Accepted: 04 December 2024 Published: 20 December 2024
MSC : 35J25, 35J40, 35B25, 65N06, 65N12, 65N15, 65N50

This work is the continuation of ^[11], where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most interesting and difficult case for this type of problem. It is well known that then, in general, overlapping regular or parabolic boundary layers appear in the solution of the continuous problem. To solve numerically the problem, the classical upwind finite difference scheme, defined on special piecewise uniform Shsihkin meshes, was used, proving its uniform convergence, with respect to all parameters, for four different ratios between them. In this paper, we complete the previous analysis, considering the two cases for these possible ratios, that were not considered in ^[11]. To see in practice the efficiency of the numerical method, we show the numerical results obtained with our algorithm for a test problem, when the cases analyzed in this work are fixed; from those results, the uniform convergence of the numerical algorithm follows, in agreement with the theoretical results.
- 2D elliptic coupled systems,
- singularly perturbed problems,
- upwind scheme,
- Shishkin meshes,
- uniform convergence
Citation: Ram Shiromani, Carmelo Clavero. An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ[J]. AIMS Mathematics, 2024, 9(12): 35570-35598. doi: 10.3934/math.20241688

Related Papers:

Abstract

This work is the continuation of ^[11], where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most interesting and difficult case for this type of problem. It is well known that then, in general, overlapping regular or parabolic boundary layers appear in the solution of the continuous problem. To solve numerically the problem, the classical upwind finite difference scheme, defined on special piecewise uniform Shsihkin meshes, was used, proving its uniform convergence, with respect to all parameters, for four different ratios between them. In this paper, we complete the previous analysis, considering the two cases for these possible ratios, that were not considered in ^[11]. To see in practice the efficiency of the numerical method, we show the numerical results obtained with our algorithm for a test problem, when the cases analyzed in this work are fixed; from those results, the uniform convergence of the numerical algorithm follows, in agreement with the theoretical results.

References

[1]	K. Aarthika, V. Shanthi, H. Ramos, A computational approach for a two-parameter singularly perturbed system of partial differential equations with discontinuous coefficients, Appl. Math. Comput., 434 (2022), 127409. https://doi.org/10.1016/j.amc.2022.127409 doi: 10.1016/j.amc.2022.127409
[2]	G. I. Barenblatt, I. P. Zheltov, L. N. Kochin, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286–1303. https://doi.org/10.1016/0021-8928(60)90107-6 doi: 10.1016/0021-8928(60)90107-6
[3]	P. Bhathawala, A. Verma, A two-parameter singularly perturbation solution of one dimension flow unstaurated prorous media, Appl. Math., 43 (1975), 380–384. https://doi.org/10.1119/1.9844 doi: 10.1119/1.9844
[4]	Z. Cen, Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, J. Syst. Sci. Complex., 18 (2005), 498–510.
[5]	J. Chen, J. R. O'Malley, On the asymptotic solution of a two-parameter boundary value problem of chemical reactor theory, SIAM J. Appl. Math., 26 (1974), 91–112. https://doi.org/10.1016/S0168-9274(98)00014-2 doi: 10.1016/S0168-9274(98)00014-2
[6]	C. Clavero, J. C. Jorge, F. Lisbona, G. I. Shishkin, A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems, Appl. Num. Math., 27 (1998), 211–231. https://doi.org/10.1016/S0168-9274(98)00014-2 doi: 10.1016/S0168-9274(98)00014-2
[7]	C. Clavero, J. C. Jorge, An efficient numerical method for singularly perturbed time dependent parabolic 2D convection-diffusion systems, J. Comput. Appl. Math., 354 (2019), 431–444. https://doi.org/10.1016/j.cam.2018.10.033 doi: 10.1016/j.cam.2018.10.033
[8]	C. Clavero, J. C. Jorge, A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems, Appl. Num. Math., 183 (2023), 317–332. https://doi.org/10.1016/j.apnum.2022.09.012 doi: 10.1016/j.apnum.2022.09.012
[9]	C. Clavero, R. Shiromani, V. Shanthi, A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2D elliptic convection-reaction-diffusion PDEs, J. Comput. Appl. Math., 434 (2024), 115422. https://doi.org/10.1016/j.cam.2023.115422 doi: 10.1016/j.cam.2023.115422
[10]	C. Clavero, R. Shiromani, V. Shanthi, A computational approach for 2D elliptic singularly perturbed weakly-coupled systems of convection-diffusion type with multiple scales and parameters in the diffusion and the convection terms, Math. Meth. Appl. Sci., (2024), 1–32. https://doi.org/10.1002/mma.10204
[11]	C. Clavero, R. Shiromani, An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, unpblished work.
[12]	P. L. Farrell, A. Hegarty, J. J. H. Miller, E. O'Riordan, G. I. Shishkin, Robust computational techniques for boundary layers, CRC Press (2000).
[13]	L. Govindarao, J. Mohapatra, S. R. Sahu, Uniformly convergent numerical method for singularly perturbed two parameter time delay parabolic problem, Int. J. Appl. Comput. Math. 5 (2019). https://doi.org/10.1007/s40819-019-0672-5
[14]	L. Govindarao, S.R. Sahu, J. Mohapatra, Uniformly convergent numerical method for singularly perturbed two parameter time delay parabolic problem with two small parameter, Iran. J. Sci. Technol. T. A, 43 (2019), 2373–2383. https://doi.org/10.1007/s40995-019-00697-2 doi: 10.1007/s40995-019-00697-2
[15]	H. Hang, R. B. Kellogg, Differentiability properties of solutions of the equation $-\varepsilon^2 \Delta u+ r u = f(x, y)$ in a square, SIAM J. Math. Anal., 21 (1990), 394–408. https://doi.org/10.1137/0521022 doi: 10.1137/0521022
[16]	Y. Kan-On, M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal., 29 (1998), 1519–1536. https://doi.org/10.1137/S0036141097318328 doi: 10.1137/S0036141097318328
[17]	O. Ladyzhenskaya, N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).
[18]	T. Linss, M. Stynes, Asymptotic analysis and Shishkin-type decomposition for an elliptic convection–diffusion problem, J. Math. Anal. Appl., 261 (2001), 604–632. https://doi.org/10.1006/jmaa.2001.7550 doi: 10.1006/jmaa.2001.7550
[19]	T. Linss, The necessity of Shishkin decompositions, Appl. Math. Lett., 14 (2001), 891–896. https://doi.org/10.1016/S0893-9659(01)00061-1 doi: 10.1016/S0893-9659(01)00061-1
[20]	T. Linss, M. Stynes, Numerical solution of systems of singularly perturbed differential equations, Comput. Meth. Appl. Math., 9 (2009), 165–191. https://doi.org/10.2478/cmam-2009-0010 doi: 10.2478/cmam-2009-0010
[21]	L. B. Liu, G. Long, Y. Zhang, Parameter uniform numerical method for a system of two coupled singularly perturbed parabolic convection-diffusion equations, Adv. Diff. Equat., 450 (2018). https://doi.org/10.1186/s13662–018–1907–13
[22]	S. Nagarajan, A parameter robust fitted mesh finite difference method for a system of two reaction-convection-diffusion equations, Appl. Num. Math., 179 (2022), 87–104. https://doi.org/10.1016/j.apnum.2022.04.017 doi: 10.1016/j.apnum.2022.04.017
[23]	E. O'Riordan, M. Pickett, G. I. Shishkin, Numerical methods for singularly perturbed elliptic problems containing two perturbation parameters, Math. Model. Anal., 11 (2006), 199–212. https://doi.org/10.3846/13926292.2006.9637313 doi: 10.3846/13926292.2006.9637313
[24]	E. O'Riordan, M. Pickett, A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem, Adv. Comput. Math., 35 (2011), 57–82. https://doi.org/10.1007/s10444-010-9164-1 doi: 10.1007/s10444-010-9164-1
[25]	S. Priyadarshana, J. Mohapatra, S. R. Pattaniak, Parameter uniform optimmal order numerical approximations for time-delayed parabolic convection diffusion problems involving two small parameters, Comput. Appl. Math., 41 (2022). https://doi.org/10.1007/s40314-022-01928-w
[26]	R. M. Priyadharshini, N. Ramanujam, A. Tamilsevan, Hybrid difference schemes for a system of singularly perturbed convection-diffusion equations, J. Appl. Math. Inform., 27 (2009), 1001–1015.
[27]	H. Schlichting, K. Gersten, Boundary layer theory, Springer (2016). https://doi.org/10.1007/978-3-662-52919-5
[28]	M. K. Singh, S. Natesan, Numerical analysis of singularly perturbed system of parabolic convection-diffusion problem with regular boundary layers, Diff. Equat. Dyn. Syst., (2019). https://doi.org/10.1007/s12591-019-00462-2
[29]	M. K. Singh, S. Natesan, A parameter-uniform hybrid finite difference schme for singularly perturbed system of parabolic convection-diffusion problems, Int. J. Comput. Math., 97 (2020), 875–903. https://doi.org/10.1080/00207160.2019.1597972 doi: 10.1080/00207160.2019.1597972
[30]	G. P. Thomas, Towards an improved turbulence model for wave-current interactions, 2nd Annual Report to EU MAST-III Project The Kinematics and Dynamics of Wave-Current Interactions (1998).

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)