Comptes Rendus
no. S1
A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations
Tomás Chacón Rebollo1  ; Antonio Domínguez-Delgado2 ; Macarena Gómez Marmol3
1 Departamento de Ecuaciones Diferenciales, Universidad de Sevilla & IMUS, c/ Tarfia, 41012 Sevilla, Spain
2 Departamento de Matemática Aplicada 1, Universidad de Sevilla, Avda.Reina Mercedes 2, 41012 Sevilla, Spain
3 Departamento de Ecuaciones Diferenciales, Universidad de Sevilla, c/ Tarfia, 41012 Sevilla, Spain
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 293-305.

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 st century” conference held in Tours in 1997 honouring the 60 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.

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DOI : 10.5802/crmeca.187
Mots clés : VMS methods, dominant convection, a posteriori filtering, stabilisation, convection-diffusion equation, traffic flow equation
Tomás Chacón Rebollo 1 ; Antonio Domínguez-Delgado 2 ; Macarena Gómez Marmol 3

1 Departamento de Ecuaciones Diferenciales, Universidad de Sevilla & IMUS, c/ Tarfia, 41012 Sevilla, Spain
2 Departamento de Matemática Aplicada 1, Universidad de Sevilla, Avda.Reina Mercedes 2, 41012 Sevilla, Spain
3 Departamento de Ecuaciones Diferenciales, Universidad de Sevilla, c/ Tarfia, 41012 Sevilla, Spain
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Comptes Rendus. M\'ecanique},
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Tomás Chacón Rebollo; Antonio Domínguez-Delgado; Macarena Gómez Marmol. A posteriori Variational Multiscale Methods for the 1D convection-diffusion equations. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 293-305. doi : 10.5802/crmeca.187. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.187/

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[5] Tomás Chacón Rebollo; Roger Lewandowski Mathematical and numerical foundations of turbulence models and applications, Modeling and Simulation in Science, Engineering and Technology, Springer, 2014 | DOI | Zbl

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