Comptes Rendus
no. 2
Partial differential equations/Numerical analysis
Uniform temporal convergence of numerical schemes for miscible flow through porous media

Presented by: the Editorial Board

Kyle S. Talbot1
1 School of Mathematical Sciences, Monash University, Victoria 3800, Australia
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 161-165.

The Hybrid Mimetic Mixed (HMM) family of schemes contains the Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods. This note proves that HMM schemes, when applied to a model of the miscible displacement of one incompressible fluid by another through a porous medium, produce approximations of the concentration variable that converge uniformly towards the exact concentration.

La famille de schémas hybrides mimétiques mixtes (HMM) englobe les méthodes volumes finis hybrides, différences finies mimétiques et volumes finis mixtes. Cette note prouve que les schémas HMM, appliqués à un modèle de déplacement incompressible miscible en milieu poreux d'un fluide par un autre, produisent des concentrations approchées qui convergent uniformément vers la concentration exacte.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.11.007

Kyle S. Talbot 1

1 School of Mathematical Sciences, Monash University, Victoria 3800, Australia 
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Kyle S. Talbot. Uniform temporal convergence of numerical schemes for miscible flow through porous media. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 161-165. doi : 10.1016/j.crma.2015.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.007/

[1] F. Brezzi; K. Lipnikov; V. Simoncini A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., Volume 15 (2005) no. 10, pp. 1533-1551

[2] C. Chainais-Hillairet; J. Droniou Convergence analysis of a mixed finite volume scheme for an elliptic–parabolic system modeling miscible fluid flows in porous media, SIAM J. Numer. Anal., Volume 45 (2007) no. 5, pp. 2228-2258 (electronic)

[3] J. Droniou; R. Eymard A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math., Volume 105 (2006) no. 1, pp. 35-71

[4] J. Droniou; R. Eymard Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations, Numer. Math. (2015) (in press) | DOI

[5] J. Droniou; R. Eymard; T. Gallouët; R. Herbin A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 2, pp. 265-295

[6] J. Droniou; R. Eymard; T. Gallouët; R. Herbin Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 13, pp. 2395-2432

[7] J. Droniou; K.S. Talbot On a miscible displacement model in porous media flow with measure data, SIAM J. Math. Anal., Volume 46 (2014) no. 5, pp. 3158-3175

[8] R. Eymard; T. Gallouët; R. Herbin Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., Volume 30 (2010) no. 4, pp. 1009-1043

[9] D.W. Peaceman Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977

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