A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations
The present work is a continuation of a paper presented by the two first authors in the proceedings of the “Computational Science for the $21^{\rm st}$ century” conference held in Tours in 1997 honouring the $60^{\rm th}$ birthday of Roland Glowinski. It is devoted to the solution of 1D convection-d...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-06-01
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| Series: | Comptes Rendus. Mécanique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.187/ |
| _version_ | 1797651361821622272 |
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| author | Chacón Rebollo, Tomás Domínguez-Delgado, Antonio Gómez Marmol, Macarena |
| author_facet | Chacón Rebollo, Tomás Domínguez-Delgado, Antonio Gómez Marmol, Macarena |
| author_sort | Chacón Rebollo, Tomás |
| collection | DOAJ |
| description | The present work is a continuation of a paper presented by the two first authors in the proceedings of the “Computational Science for the $21^{\rm st}$ century” conference held in Tours in 1997 honouring the $60^{\rm th}$ birthday of Roland Glowinski. It is devoted to the solution of 1D convection-diffusion equations in dominant convection regime situations. In that paper, an “a posteriori” VMS filtering technique was introduced. We present an extension of this technique to nonlinear convection-diffusion equations (a traffic model), providing an efficient method for the resolution of shocks from just the Galerkin solution at targeted times. We also present a residual-based “a posteriori” VMS filtering, that provides quite accurate stable solutions, can be extended to multi-dimensional problems, and can be applied locally. We finally present some numerical tests exhibiting the high accuracy of the obtained solutions. |
| first_indexed | 2024-03-11T16:14:53Z |
| format | Article |
| id | doaj.art-17257b79f6314fc3a17efdd6df35f8b8 |
| institution | Directory Open Access Journal |
| issn | 1873-7234 |
| language | English |
| last_indexed | 2024-03-11T16:14:53Z |
| publishDate | 2023-06-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mécanique |
| spelling | doaj.art-17257b79f6314fc3a17efdd6df35f8b82023-10-24T14:21:34ZengAcadémie des sciencesComptes Rendus. Mécanique1873-72342023-06-0111310.5802/crmeca.18710.5802/crmeca.187A posteriori Variational Multiscale Methods for the 1D convection-diffusion equationsChacón Rebollo, Tomás0Domínguez-Delgado, Antonio1Gómez Marmol, Macarena2Departamento de Ecuaciones Diferenciales, Universidad de Sevilla & IMUS, c/ Tarfia, 41012 Sevilla, SpainDepartamento de Matemática Aplicada 1, Universidad de Sevilla, Avda.Reina Mercedes 2, 41012 Sevilla, SpainDepartamento de Ecuaciones Diferenciales, Universidad de Sevilla, c/ Tarfia, 41012 Sevilla, SpainThe present work is a continuation of a paper presented by the two first authors in the proceedings of the “Computational Science for the $21^{\rm st}$ century” conference held in Tours in 1997 honouring the $60^{\rm th}$ birthday of Roland Glowinski. It is devoted to the solution of 1D convection-diffusion equations in dominant convection regime situations. In that paper, an “a posteriori” VMS filtering technique was introduced. We present an extension of this technique to nonlinear convection-diffusion equations (a traffic model), providing an efficient method for the resolution of shocks from just the Galerkin solution at targeted times. We also present a residual-based “a posteriori” VMS filtering, that provides quite accurate stable solutions, can be extended to multi-dimensional problems, and can be applied locally. We finally present some numerical tests exhibiting the high accuracy of the obtained solutions.https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.187/VMS methodsdominant convectiona posteriori filteringstabilisationconvection-diffusion equationtraffic flow equation |
| spellingShingle | Chacón Rebollo, Tomás Domínguez-Delgado, Antonio Gómez Marmol, Macarena A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations Comptes Rendus. Mécanique VMS methods dominant convection a posteriori filtering stabilisation convection-diffusion equation traffic flow equation |
| title | A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations |
| title_full | A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations |
| title_fullStr | A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations |
| title_full_unstemmed | A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations |
| title_short | A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations |
| title_sort | posteriori variational multiscale methods for the 1d convection diffusion equations |
| topic | VMS methods dominant convection a posteriori filtering stabilisation convection-diffusion equation traffic flow equation |
| url | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.187/ |
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