Computational Geosciences

, Volume 18, Issue 3–4, pp 285–296 | Cite as

TP or not TP, that is the question

  • R. EymardEmail author
  • T. Gallouët
  • C. Guichard
  • R. Herbin
  • R. Masson


We give here a comparative study on the mathematical analysis of two (classes of) discretization schemes for the computation of approximate solutions to incompressible two-phase flow problems in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The second class contains the so-called approximate gradient schemes, which include finite elements with mass lumping, mixed finite elements, and mimetic finite differences. Both (classes of) schemes are nonconforming and can be expressed using discrete function and gradient reconstructions within a variational formulation. Each class has its specific advantages and drawbacks: monotony properties are natural with the two-point finite volume scheme, but meshes are restricted due to consistency issues; on the contrary, gradient schemes can be used on general meshes, but monotony properties are difficult to obtain.


Two-phase flow in porous media Two-point flux approximation Finite volume scheme Gradient scheme 


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  1. 1.
    Aavatsmark, I., Barkve, T., Boe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: derivation of the methods. SIAM J. Sci. Comp. 19, 1700–1716 (1998)CrossRefGoogle Scholar
  2. 2.
    Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)CrossRefGoogle Scholar
  3. 3.
    Brenner, K.: Hybrid finite volume scheme for a two-phase flow in heterogeneous porous media. ESAIM: Proc. 35, 210–215 (2012).
  4. 4.
    Brenner, K., Cances, C., Hilhorst, D.: A convergent finite volume scheme for two-phase flows in porous media with discontinuous capillary pressure field:Finite Volumes for Complex Applications VI Problems & Perspectives, pp. 185–193. Springer, Berlin (2011)CrossRefGoogle Scholar
  5. 5.
    Brenner, K., Masson, R.: Convergence of a vertex centred discretization of two-phase Darcy flows on general meshes. Int. J. Finite 10, 1–37 (2013) Google Scholar
  6. 6.
    Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation. Elsevier, Amsterdam (1986)Google Scholar
  7. 7.
    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 Methods Appl. Sci. (M3AS) 23(13), 2395–2432 (2013) CrossRefGoogle Scholar
  8. 8.
    Droniou, J., Gallouët, T., Herbin, R.: A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal. 41(6), 1997–2031 (2003). doi: 10.1137/S0036142902405205 CrossRefGoogle Scholar
  9. 9.
    Eymard, R., Féron, P., Gallouet, T., Herbin, R., Guichard, C.: Gradient schemes for the Stefan problem. Int. J Finite Volumes 10 special (2013)
  10. 10.
    Eymard, R., Gallouët, T., Herbin, R.: Techniques of Scientific Computing. Part III, Handbook of Numerical Analysis Ciarlet, P.G., Lions, J.L. (eds.), Vol. VII. North-Holland, Amsterdam (2000)Google Scholar
  11. 11.
    Eymard, R., Gallouët, T., Herbin, R., Michel, A.: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92(1), 41–82 (2002)CrossRefGoogle Scholar
  12. 12.
    Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: Math. Model. Numer. Anal. 46(02), 265–290 (2012)CrossRefGoogle Scholar
  13. 13.
    Eymard, R., Henry, G., Herbin, R., Hubert, F., Klöfkorn, R., Manzini, G.: 3d benchmark on discretization schemes for anisotropic diffusion problems on general grids. Finite Vol. Complex Appl. VI Probl. Perspect. 895–930 (2011)Google Scholar
  14. 14.
    Eymard, R., Herbin, R., Latché, J.C.: Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes. SIAM J. Numer. Anal. 45(1), 1–36 (2007)CrossRefGoogle Scholar
  15. 15.
    Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme. M2AN Math. Model. Numer. Anal. 37(6), 937–972 (2003)CrossRefGoogle Scholar
  16. 16.
    Eymard, R., Schleper, V.: Study of a numerical scheme for miscible two-phase flow in porous media. Numer. Methods Partial Differ. Equ. (2013). doi: 10.1002/num.21823
  17. 17.
    Gallouët, T., Herbin, R.: Convergence of linear finite elements for diffusion equations with measure data. Comptes Rendus Mathematique 338(1), 81–84 (2004)CrossRefGoogle Scholar
  18. 18.
    Herbin, R., Hubert, F., et al.: Finite Volume Complex Applications V, pp. 659–692 (2008)Google Scholar
  19. 19.
    Le Potier, C.: A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. Int. J. Finite 6(2), 20 (2009)Google Scholar
  20. 20.
    Le Potier, C., Mahamane, A.: A nonlinear correction and maximum principle for diffusion operators with hybrid schemes. CR Acad. Sci. Paris, Ser. I 350, 101–106 (2012)CrossRefGoogle Scholar
  21. 21.
    Michel, A.: A finite volume scheme for two-phase immiscible flow in porous media. SIAM J. Numer. Anal. 41(4), 1301–1317 (2003). doi: 10.1137/S0036142900382739 CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • R. Eymard
    • 1
    Email author
  • T. Gallouët
    • 2
  • C. Guichard
    • 3
  • R. Herbin
    • 2
  • R. Masson
    • 3
  1. 1.L.A.M.A., UMR 8050Université Paris-EstParisFrance
  2. 2.UMR 7353, LATP, Centrale Marseille, CNRSAix-Marseille UniversitéMarseilleFrance
  3. 3.Team Coffee INRIA Sophia Antipolis Méditerranée, UMR 7351, LJAD, CNRSUniversity Nice Sophia AntipolisNiceFrance

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