Advertisement

A Gradient Scheme for the Discretization of Richards Equation

  • Konstantin Brenner
  • Danielle Hilhorst
  • Huy Cuong Vu Do
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 78)

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.

Keywords

Relative Permeability Dirichlet Boundary Condition Adaptive Mesh Absolute Permeability Richards Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

D. Hilhorst and H.C. Vu Do acknowledge the support of the ITN Marie Curie Project FIRST and of the Fondation Jacques Hadamard

References

  1. 1.
    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 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    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
  4. 4.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Eymard, R., Gutnic, M., Hilhorst, D.: The finite volume method for Richards equation. Comput. Geosci. 3(3–4), 259–294 (2000)MathSciNetGoogle Scholar
  6. 6.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Sochala, P.: Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement. Thèse de doctorat, Ecole Nationale des Ponts et Chaussées (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Konstantin Brenner
    • 1
    • 2
  • Danielle Hilhorst
    • 3
  • Huy Cuong Vu Do
    • 4
  1. 1.LJADUniversity Nice Sophia-AntipolisNiceFrance
  2. 2.Coffee Team Inria Sophia-Antipolis-MéditerranéeValbonneFrance
  3. 3.Laboratoire de MathématiquesCNRS et Université de Paris-SudOrsayFrance
  4. 4.Laboratoire de MathématiquesUniversité de Paris-SudOrsayFrance

Personalised recommendations