Abstract
This work is devoted to a Finite Volume method to approximate the solution of a convection-diffusion equation involving a Joule term. We propose a way to discretize this so-called “Joule effect” term in a consistent way with the non linear diffusion one, in order to ensure some maximum principle properties on the solution. We then investigate the numerical behavior of the scheme on two original benchmarks.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bradji, A., Herbin, R.: Discretization of coupled heat and electrical diffusion problems by finite-element and finite-volume methods. IMA J. Numer. Anal. 28(3), 469–495 (2008). https://doi.org/10.1093/imanum/drm030
Calgaro, C., Colin, C., Creusé, E.: A combined finite volume - finite element scheme for a low-Mach system involving a Joule term. AIMS Math. 5(1), 311–331 (2020)
Calgaro, C., Colin, C., Creusé, E., Zahrouni, E.: Approximation by an iterative method of a low-Mach model with temperature dependant viscosity. Math. Methods Appl. Sci. 42, 250–271 (2019)
Chainais-Hillairet, C.: Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. Internat. J. Numer. Methods Fluids 59(3), 239–257 (2009). https://doi.org/10.1002/fld.1393
Chainais-Hillairet, C., Peng, Y.J., Violet, I.: Numerical solutions of Euler-Poisson systems for potential flows. Appl. Numer. Math. 59(2), 301–315 (2009). https://doi.org/10.1016/j.apnum.2008.02.006
Colin, C.: Analyse et simulation numérique par méthode combinée volumes finis—eléments finis de modèles de type faible mach. Ph.D. thesis, Université de Lille (2019)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, pp. 713–1020. North-Holland, Amsterdam (2000)
Eymard, R., Gallouët, T., Herbin, R.: A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal. 26(2), 326–353 (2006). https://doi.org/10.1093/imanum/dri036
Huang, F., Tan, W.: On the strong solution of the ghost effect system. SIAM J. Math. Anal. 49(5), 3496–3526 (2017). https://doi.org/10.1137/16M106964X
Levermore, C., Sun, W., Trivisa, K.: Local well-posedness of a ghost system effect. Indiana Univ. Math. J. 60, 517–576 (2011)
Acknowledgements
This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Calgaro, C., Creusé, E. (2020). A Finite Volume Method for a Convection-Diffusion Equation Involving a Joule Term. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_37
Download citation
DOI: https://doi.org/10.1007/978-3-030-43651-3_37
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43650-6
Online ISBN: 978-3-030-43651-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)