Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

Jérôme Droniou, Robert Eymard, Alain Prignet, Kyle S. Talbot

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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 (Formula presented.) 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.

Original languageEnglish
Pages (from-to)333-374
Number of pages42
JournalFoundations of Computational Mathematics
Volume19
Issue number2
DOIs
Publication statusPublished - 2019

Keywords

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

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