Abstract
We propose a finite volume method on general meshes for the discretization of Richards equation, an elliptic—parabolic equation modeling groundwater flow. The diffusion term, which can be anisotropic and heterogeneous, is discretized in a gradient scheme framework, which can be applied to a wide range of unstructured possibly non-matching polyhedral meshes in arbitrary space dimension. More precisely, we implement the SUSHI scheme which is also locally conservative. As is needed for Richards equation, the time discretization is fully implicit. We obtain a convergence result based upon energy-type estimates and the application of the Fréchet-Kolmogorov compactness theorem. We implement the scheme and present the results of a number of numerical tests.
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References
Angelini, O., Brenner, K., Hilhorst, D.: A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation. Numer. Math. 123, 219–257 (2013). doi:10.1007/s00211-012-0485-5
Droniou, J., Eymard, R., Gallouet, T., Herbin, R.: Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Meth. Appl. Sci. 23(13), 2395–2432 (2013)
Eymard, R., Guichard, C., Herbin, R., Masson, R.: Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. ZAMM (2013). doi:10.1002/zamm.201200206
Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. Sushi: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010)
Eymard, R., Gutnic, M., Hilhorst, D.: The finite volume method for Richards equation. Comput. Geosci. 3(3–4), 259–294 (2000)
Eymard, R., Hilhorst, D., Vohralik, M.: A combined finite volume scheme nonconforming/ mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105, 73–131 (2006)
Haverkamp, R., Vauclin, M., Touma, J., Wierenga, P., Vachaud, G.: A comparison of numerical simulation models for one-dimensional infiltration. Soil Sci. Soc. Am. J. 41(2),285–294 (1977)
Sochala, P.: Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement. Thèse de doctorat, Ecole Nationale des Ponts et Chaussées (2008)
Acknowledgments
D. Hilhorst and H.C. Vu Do acknowledge the support of the ITN Marie Curie Project FIRST and of the Fondation Jacques Hadamard
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Brenner, K., Hilhorst, D., Vu Do, H.C. (2014). A Gradient Scheme for the Discretization of Richards Equation. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems. Springer Proceedings in Mathematics & Statistics, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-319-05591-6_53
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DOI: https://doi.org/10.1007/978-3-319-05591-6_53
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