A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems
We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform converg...
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| Published in | Journal of applied mathematics & computing Vol. 70; no. 6; pp. 5645 - 5668 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2024
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 1598-5865 1865-2085 |
| DOI | 10.1007/s12190-024-02178-1 |
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| Abstract | We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper. |
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| AbstractList | We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper. |
| Author | Kumar, Shashikant Kuldeep Ramos, Higinio Kumar, Sunil |
| Author_xml | – sequence: 1 givenname: Shashikant surname: Kumar fullname: Kumar, Shashikant organization: Department of Mathematics, Indian Institute of Technology Delhi – sequence: 2 givenname: Sunil surname: Kumar fullname: Kumar, Sunil organization: Department of Mathematical Sciences, Indian Institute of Technology (BHU) – sequence: 3 givenname: Higinio orcidid: 0000-0003-2791-6230 surname: Ramos fullname: Ramos, Higinio email: higra@usal.es organization: Scientific Computing Group, Universidad de Salamanca, Escuela Politécnica Superior de Zamora, Campus Viriato – sequence: 4 surname: Kuldeep fullname: Kuldeep organization: Department of Mathematical Sciences, Indian Institute of Technology (BHU) |
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| Cites_doi | 10.1016/j.matcom.2019.03.007 10.1553/etna_vol51s315 10.1016/j.aml.2023.108755 10.1007/s40314-022-02090-z 10.1016/j.camwa.2014.09.004 10.1016/j.cam.2020.113273 10.30755/NSJOM.07880 10.1002/num.20574 10.1016/j.apm.2011.10.012 10.1007/s40314-014-0171-6 10.1016/S0377-0427(02)00861-0 10.1016/S0168-9274(98)00014-2 10.1016/j.cnsns.2010.07.020 10.1007/s40819-017-0380-y 10.1002/num.22420 10.1142/2933 10.2478/cmam-2003-0023 10.1002/num.22930 10.1201/9781482285727 10.1016/j.matcom.2018.12.010 10.2989/16073606.2014.981708 10.1007/s00211-007-0083-0 |
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| Keywords | Uniform convergence 65M15 Singular perturbation 65M06 Bakhvalov mesh 65M55 65M12 Upwind scheme Degenerate parabolic problem |
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| References | Munyakazi, Patidar (CR21) 2015; 38 Linß (CR16) 2009 Zhang, Liu (CR30) 2023; 39 Mbroh, Munyakazi (CR18) 2019; 165 Kadalbajoo, Gupta (CR11) 2010; 217 Munyakazi, Patidar, Sayi (CR22) 2019; 35 Gupta, Kadalbajoo (CR9) 2011; 27 Andreev, Kopteva (CR1) 1999; 34 Nhan, Vulanovic (CR24) 2018; 48 Gupta, Kadalbajoo (CR10) 2011; 16 Liao, Liu, Ye, Liu (CR15) 2023; 145 Bakhvalov (CR3) 1969; 9 Clavero, Jorge, Lisbona, Shishkin (CR6) 1998; 27 Munyakazi (CR20) 2015; 34 Kumar, Kumar (CR13) 2014; 68 Toprakseven (CR28) 2022; 41 Munyakazi, Patidar, Sayi (CR23) 2019; 160 Roos, Stynes, Tobiska (CR27) 2008 Miller, O’Riordan, Shishkin (CR19) 1996 Rao, Kumar (CR26) 2012; 219 Andreev, Savin (CR2) 1995; 35 Farrell, Hegarty, Miller, O’Riordan, Shishkin (CR8) 2000 Clavero, Jorge, Lisbona (CR5) 2003; 154 Vulanovic (CR29) 1983; 13 Kumar, Kumar (CR12) 2012; 36 Majumdar, Natesan (CR17) 2017; 3 Bujanda, Clavero, Gracia, Jorge (CR4) 2007; 107 Dunne, O’Riordan, Shishkin (CR7) 2003; 3 Kumar, Vigo-Aguiar (CR14) 2022; 404 Nhan, Vulanovic (CR25) 2019; 51 JB Munyakazi (2178_CR21) 2015; 38 V Gupta (2178_CR10) 2011; 16 VB Andreev (2178_CR1) 1999; 34 C Clavero (2178_CR6) 1998; 27 SCS Rao (2178_CR26) 2012; 219 NA Mbroh (2178_CR18) 2019; 165 S Kumar (2178_CR14) 2022; 404 M Kumar (2178_CR12) 2012; 36 B Bujanda (2178_CR4) 2007; 107 V Gupta (2178_CR9) 2011; 27 S Kumar (2178_CR13) 2014; 68 JJH Miller (2178_CR19) 1996 MK Kadalbajoo (2178_CR11) 2010; 217 TA Nhan (2178_CR24) 2018; 48 VB Andreev (2178_CR2) 1995; 35 HG Roos (2178_CR27) 2008 JB Munyakazi (2178_CR20) 2015; 34 JB Munyakazi (2178_CR22) 2019; 35 JB Munyakazi (2178_CR23) 2019; 160 Y Liao (2178_CR15) 2023; 145 T Linß (2178_CR16) 2009 R Vulanovic (2178_CR29) 1983; 13 TA Nhan (2178_CR25) 2019; 51 P Farrell (2178_CR8) 2000 RK Dunne (2178_CR7) 2003; 3 Suayip Toprakseven (2178_CR28) 2022; 41 J Zhang (2178_CR30) 2023; 39 C Clavero (2178_CR5) 2003; 154 A Majumdar (2178_CR17) 2017; 3 NS Bakhvalov (2178_CR3) 1969; 9 |
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| SubjectTerms | Boundary layers Computational Mathematics and Numerical Analysis Convection Convergence Diffusion barriers Diffusion layers Error analysis Mathematical and Computational Engineering Mathematics Mathematics and Statistics Mathematics of Computing Numerical methods Original Research Singular perturbation Theory of Computation Truncation errors |
| Title | A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems |
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