Numerical Analysis of Singularly Perturbed System of Parabolic Convection–Diffusion Problem with Regular Boundary Layers
In this article, we obtain the numerical solution of singularly perturbed system of parabolic convection–diffusion problems exhibiting boundary layer. The proposed numerical scheme consists of the backward-Euler method for the time derivative and an upwind finite difference scheme for the spatial de...
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| Published in | Differential equations and dynamical systems Vol. 30; no. 3; pp. 695 - 717 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New Delhi
Springer India
01.07.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0971-3514 0974-6870 |
| DOI | 10.1007/s12591-019-00462-2 |
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| Abstract | In this article, we obtain the numerical solution of singularly perturbed system of parabolic convection–diffusion problems exhibiting boundary layer. The proposed numerical scheme consists of the backward-Euler method for the time derivative and an upwind finite difference scheme for the spatial derivatives. We analyze the scheme on a piecewise-uniform Shishkin mesh for the spatial discretization to establish uniform convergence with respect to the perturbation parameters. For the proposed scheme, the stability analysis is presented and parameter-uniform error estimate is derived. In order to validate the theoretical results, we have carried out some numerical experiments. |
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| AbstractList | In this article, we obtain the numerical solution of singularly perturbed system of parabolic convection–diffusion problems exhibiting boundary layer. The proposed numerical scheme consists of the backward-Euler method for the time derivative and an upwind finite difference scheme for the spatial derivatives. We analyze the scheme on a piecewise-uniform Shishkin mesh for the spatial discretization to establish uniform convergence with respect to the perturbation parameters. For the proposed scheme, the stability analysis is presented and parameter-uniform error estimate is derived. In order to validate the theoretical results, we have carried out some numerical experiments. |
| Author | Natesan, Srinivasan Singh, Maneesh Kumar |
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| Cites_doi | 10.1080/0020716042000301798 10.4208/cicp.scpde14.21s 10.1090/mmono/023 10.1007/s10444-007-9058-z 10.1007/978-1-4612-1114-3 10.1142/8410 10.1201/9781482285727 10.1016/j.apnum.2004.05.006 10.1007/978-3-642-05134-0 10.1007/s10915-013-9814-9 10.1007/s00607-006-0215-x 10.1016/0377-0427(88)90315-9 |
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| Keywords | Finite difference scheme Uniform convergence Singularly perturbed system Shishkin mesh AMS 65M06 CR G1.8 Parabolic convection–diffusion problems 65M12 Boundary layers |
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| References | Ng-Stynes, O’Riordan, Stynes (CR15) 1988; 21 Deb, Natesan (CR5) 2008; 38 Farrell, Hegarty, Miller, O’Riordan, Shishkin (CR7) 2000 Miller, O’Riordan, Shishkin (CR13) 2012 Doolan, Miller, Schilders (CR6) 1980 Ladyzenskaja, Solonnikov, Ural’ceva (CR9) 1968 Linß (CR11) 2007; 79 Chang, Howes (CR3) 1984 Bellew, O’Riordan (CR1) 2004; 51 Pao (CR17) 1992 Liu, Chen (CR12) 2014; 61 Hsieh, Shih, Yang, You (CR8) 2016; 19 Morton (CR14) 1996 O’Riordan, Stynes (CR16) 2009; 30 Das, Natesan (CR4) 2013; 10 Linß (CR10) 2010 Cen (CR2) 2005; 82 T Linß (462_CR11) 2007; 79 CV Pao (462_CR17) 1992 S Bellew (462_CR1) 2004; 51 T Linß (462_CR10) 2010 KW Chang (462_CR3) 1984 OA Ladyzenskaja (462_CR9) 1968 P Das (462_CR4) 2013; 10 E O’Riordan (462_CR16) 2009; 30 LB Liu (462_CR12) 2014; 61 JJH Miller (462_CR13) 2012 KW Morton (462_CR14) 1996 Z Cen (462_CR2) 2005; 82 MJ Ng-Stynes (462_CR15) 1988; 21 P-W Hsieh (462_CR8) 2016; 19 EP Doolan (462_CR6) 1980 PA Farrell (462_CR7) 2000 BS Deb (462_CR5) 2008; 38 |
| References_xml | – year: 1992 ident: CR17 publication-title: Nonlinear Parabolic and Elliptic Equations – volume: 82 start-page: 177 issue: 2 year: 2005 end-page: 192 ident: CR2 article-title: Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations publication-title: Int. J. Comput. Math. doi: 10.1080/0020716042000301798 – volume: 19 start-page: 1287 issue: 5 year: 2016 end-page: 1301 ident: CR8 article-title: A novel technique for constructing difference schemes for systems of singularly perturbed equations publication-title: Commun. Comput. Phys. doi: 10.4208/cicp.scpde14.21s – year: 1968 ident: CR9 publication-title: Linear and Quasi-Linear Equations of Parabolic Type doi: 10.1090/mmono/023 – volume: 30 start-page: 101 issue: 2 year: 2009 end-page: 121 ident: CR16 article-title: Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems publication-title: Adv. Comput. Math. doi: 10.1007/s10444-007-9058-z – year: 1984 ident: CR3 publication-title: Nonlinear Singular Perturbation Phenomena: Theory and Applications, Applied Mathematical Sciences doi: 10.1007/978-1-4612-1114-3 – volume: 38 start-page: 179 issue: 2 year: 2008 end-page: 200 ident: CR5 article-title: Richardson extrapolation mehtod for singularly perturbed coupled system of convection-diffusion boundary-value problems publication-title: CMES Comput. Model. Eng. Sci. – volume: 10 start-page: 17 issue: 1 year: 2013 ident: CR4 article-title: Numerical solution of a system of singularly perturbed convection-diffusion boundary-value problems using mesh equidistribution technique publication-title: Aust. J. Math. Anal. 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Math. doi: 10.1016/0377-0427(88)90315-9 – volume-title: Layer-adapted Meshes for Reaction-Convection-Diffusion Problems year: 2010 ident: 462_CR10 doi: 10.1007/978-3-642-05134-0 – volume-title: Numerical Solution of Convection-Diffusion Problems, Applied Mathematics and Mathematical Computation year: 1996 ident: 462_CR14 – volume: 38 start-page: 179 issue: 2 year: 2008 ident: 462_CR5 publication-title: CMES Comput. Model. Eng. Sci. – volume-title: Robust Computational Techniques for Boundary Layers year: 2000 ident: 462_CR7 doi: 10.1201/9781482285727 – volume: 51 start-page: 171 issue: 2–3 year: 2004 ident: 462_CR1 publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2004.05.006 – volume-title: Linear and Quasi-Linear Equations of Parabolic Type year: 1968 ident: 462_CR9 doi: 10.1090/mmono/023 – volume: 30 start-page: 101 issue: 2 year: 2009 ident: 462_CR16 publication-title: Adv. Comput. Math. doi: 10.1007/s10444-007-9058-z – volume: 19 start-page: 1287 issue: 5 year: 2016 ident: 462_CR8 publication-title: Commun. Comput. Phys. doi: 10.4208/cicp.scpde14.21s – volume: 61 start-page: 1 issue: 1 year: 2014 ident: 462_CR12 publication-title: J. Sci. Comput. doi: 10.1007/s10915-013-9814-9 – volume-title: Uniform Numerical Methods for Problems with Initial and Boundary Layers year: 1980 ident: 462_CR6 – volume: 79 start-page: 23 year: 2007 ident: 462_CR11 publication-title: Computing doi: 10.1007/s00607-006-0215-x – volume: 21 start-page: 289 issue: 3 year: 1988 ident: 462_CR15 publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(88)90315-9 – volume-title: Nonlinear Parabolic and Elliptic Equations year: 1992 ident: 462_CR17 – volume-title: Nonlinear Singular Perturbation Phenomena: Theory and Applications, Applied Mathematical Sciences year: 1984 ident: 462_CR3 doi: 10.1007/978-1-4612-1114-3 – volume: 10 start-page: 17 issue: 1 year: 2013 ident: 462_CR4 publication-title: Aust. J. Math. Anal. Appl. – volume: 82 start-page: 177 issue: 2 year: 2005 ident: 462_CR2 publication-title: Int. J. Comput. Math. doi: 10.1080/0020716042000301798 – volume-title: Fitted Numerical Methods for Singular Perturbation Problems year: 2012 ident: 462_CR13 doi: 10.1142/8410 |
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| SubjectTerms | Boundary layers Computer Science Convection Diffusion layers Engineering Error analysis Finite difference method Mathematics Mathematics and Statistics Numerical analysis Original Research Parameters Perturbation Stability analysis |
| Title | Numerical Analysis of Singularly Perturbed System of Parabolic Convection–Diffusion Problem with Regular Boundary Layers |
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