Foundations of Computational Mathematics

, Volume 19, Issue 2, pp 333–374 | Cite as

Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

  • Jérôme DroniouEmail author
  • Robert Eymard
  • Alain Prignet
  • Kyle S.  Talbot


This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in \(L^\infty (0,T; L^2(\Omega ))\) of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes is compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion.


Miscible fluid flow Coupled elliptic–parabolic problem Convergence analysis Uniform-in-time convergence Gradient discretisation method Finite differences Mass-lumped finite elements 

Mathematics Subject Classification

65M06 65M08 65M12 65M60 


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Copyright information

© SFoCM 2018

Authors and Affiliations

  • Jérôme Droniou
    • 1
    Email author
  • Robert Eymard
    • 2
  • Alain Prignet
    • 2
  • Kyle S.  Talbot
    • 1
  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité Paris-Est, UMR 8050Marne-la-Vallée Cedex 2France

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