Robin boundary value problems for a singularly perturbed weakly coupled system of convection–diffusion equations having discontinuous source term
In this paper, we consider a weakly coupled system of convection–diffusion equations subject to Robin boundary conditions, and having boundary and interior layers. The diffusion term of each equation is multiplied by a small singular perturbation parameter, but these parameters are assumed to be dif...
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| Published in | The Journal of Analysis Vol. 28; no. 1; pp. 305 - 321 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Springer Singapore
01.03.2020
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0971-3611 2367-2501 |
| DOI | 10.1007/s41478-019-00172-6 |
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| Abstract | In this paper, we consider a weakly coupled system of convection–diffusion equations subject to Robin boundary conditions, and having boundary and interior layers. The diffusion term of each equation is multiplied by a small singular perturbation parameter, but these parameters are assumed to be different in magnitude, and the source term is having a discontinuity at a point in the interior of the domain. An upwind scheme is used for the considered problem in conjunction with piecewise uniform Shishkin mesh. It is proved that the numerical approximations produced by this method are almost first order uniformly convergent with respect to both small parameters. Numerical results are presented to validate the theoretical results. |
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| AbstractList | In this paper, we consider a weakly coupled system of convection–diffusion equations subject to Robin boundary conditions, and having boundary and interior layers. The diffusion term of each equation is multiplied by a small singular perturbation parameter, but these parameters are assumed to be different in magnitude, and the source term is having a discontinuity at a point in the interior of the domain. An upwind scheme is used for the considered problem in conjunction with piecewise uniform Shishkin mesh. It is proved that the numerical approximations produced by this method are almost first order uniformly convergent with respect to both small parameters. Numerical results are presented to validate the theoretical results. |
| Author | Chawla, Sheetal Rao, S. Chandra Sekhara |
| Author_xml | – sequence: 1 givenname: S. Chandra Sekhara surname: Rao fullname: Rao, S. Chandra Sekhara email: scsr@maths.iitd.ac.in organization: Department of Mathematics, Indian Institute of Technology Delhi – sequence: 2 givenname: Sheetal surname: Chawla fullname: Chawla, Sheetal organization: Department of Mathematics, Indian Institute of Technology Delhi |
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| Cites_doi | 10.1016/j.cam.2006.02.025 10.1016/S0377-0427(02)00913-5 10.14317/jami.2015.635 10.1007/978-81-322-2485-3_60 10.1007/s12190-012-0611-7 10.1080/0020716042000301798 10.1016/j.apnum.2018.01.006 |
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| Copyright | Forum D'Analystes, Chennai 2019 |
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| Keywords | Finite difference scheme 65M15 Shishkin mesh Uniformly convergent 65M06 Singular perturbation problems Weakly coupled system Overlapping boundary layers Interior layers Discontinuous source term Convection–diffusion 65M12 |
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| References | Ansari, Hegarty (CR1) 2003; 156 Natesan, Bawa (CR6) 2007; 2 Rao, Chawla (CR8) 2015; 143 CR4 Chawla, Rao (CR3) 2015; 33 Tamilselvan, Ramanujam, Shanthi (CR12) 2007; 202 Das, Natesan (CR5) 2013; 41 CR7 Rao, Chawla (CR10) 2018; 127 Rao, Chawla (CR9) 2015; 108 Tamilselvan, Ramanujam (CR11) 2009; 27 Cen (CR2) 2005; 82 AR Ansari (172_CR1) 2003; 156 SCS Rao (172_CR9) 2015; 108 SCS Rao (172_CR10) 2018; 127 Z Cen (172_CR2) 2005; 82 A Tamilselvan (172_CR12) 2007; 202 S Chawla (172_CR3) 2015; 33 172_CR4 172_CR7 P Das (172_CR5) 2013; 41 S Natesan (172_CR6) 2007; 2 SCS Rao (172_CR8) 2015; 143 A Tamilselvan (172_CR11) 2009; 27 |
| References_xml | – volume: 202 start-page: 203 year: 2007 end-page: 216 ident: CR12 article-title: A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/j.cam.2006.02.025 – volume: 27 start-page: 1279 issue: 5–6 year: 2009 end-page: 1292 ident: CR11 article-title: A numerical method for singularly perturbed system of second order ordinary differential equations of convection diffusion type with a discontinuous source term publication-title: Journal of Applied Mathematics and Informatics – volume: 156 start-page: 221 issue: 5 year: 2003 end-page: 238 ident: CR1 article-title: Numerical solution of a convection diffusion problem with Robin boundary conditions publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/S0377-0427(02)00913-5 – volume: 33 start-page: 635 issue: 5–6 year: 2015 end-page: 648 ident: CR3 article-title: A uniformly convergent numerical method for a weakly coupled system of singularly perturbed convection–diffusion problems with boundary and weak interior layers publication-title: Journal of Applied Mathematics and Informatics doi: 10.14317/jami.2015.635 – volume: 143 start-page: 753 year: 2015 end-page: 764 ident: CR8 article-title: Numerical solution for a coupled system of singularly perturbed initial value problems with discontinuous source term publication-title: Springer Proceedings in Mathematics and Statistics doi: 10.1007/978-81-322-2485-3_60 – ident: CR7 – ident: CR4 – volume: 41 start-page: 447 year: 2013 end-page: 471 ident: CR5 article-title: A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary value problems publication-title: Journal of Applied Mathematics and Computing doi: 10.1007/s12190-012-0611-7 – 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: International Journal of Computer Mathematics doi: 10.1080/0020716042000301798 – volume: 108 start-page: 233 year: 2015 end-page: 244 ident: CR9 article-title: Second order uniformly convergent numerical method for a coupled system of singularly perturbed reaction-diffusion problems with discontinuous source term publication-title: LNCSE – volume: 127 start-page: 249 year: 2018 end-page: 265 ident: CR10 article-title: Numerical solution of singularly perturbed linear parabolic system with discontinuous source term publication-title: Applied Numerical Mathematics doi: 10.1016/j.apnum.2018.01.006 – volume: 2 start-page: 177 issue: 3–4 year: 2007 end-page: 192 ident: CR6 article-title: Second-order numerical scheme for singularly perturbed reaction-diffusion Robin problems publication-title: Journal of Numerical Analysis, Industrial and Applied Mathematics – ident: 172_CR4 – volume: 2 start-page: 177 issue: 3–4 year: 2007 ident: 172_CR6 publication-title: Journal of Numerical Analysis, Industrial and Applied Mathematics – volume: 127 start-page: 249 year: 2018 ident: 172_CR10 publication-title: Applied Numerical Mathematics doi: 10.1016/j.apnum.2018.01.006 – ident: 172_CR7 – volume: 143 start-page: 753 year: 2015 ident: 172_CR8 publication-title: Springer Proceedings in Mathematics and Statistics doi: 10.1007/978-81-322-2485-3_60 – volume: 33 start-page: 635 issue: 5–6 year: 2015 ident: 172_CR3 publication-title: Journal of Applied Mathematics and Informatics doi: 10.14317/jami.2015.635 – volume: 41 start-page: 447 year: 2013 ident: 172_CR5 publication-title: Journal of Applied Mathematics and Computing doi: 10.1007/s12190-012-0611-7 – volume: 108 start-page: 233 year: 2015 ident: 172_CR9 publication-title: LNCSE – volume: 82 start-page: 177 issue: 2 year: 2005 ident: 172_CR2 publication-title: International Journal of Computer Mathematics doi: 10.1080/0020716042000301798 – volume: 27 start-page: 1279 issue: 5–6 year: 2009 ident: 172_CR11 publication-title: Journal of Applied Mathematics and Informatics – volume: 156 start-page: 221 issue: 5 year: 2003 ident: 172_CR1 publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/S0377-0427(02)00913-5 – volume: 202 start-page: 203 year: 2007 ident: 172_CR12 publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/j.cam.2006.02.025 |
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| Title | Robin boundary value problems for a singularly perturbed weakly coupled system of convection–diffusion equations having discontinuous source term |
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