Unified convergence analysis of numerical schemes for a miscible displacement problem

TL;DR

This paper provides a unified convergence analysis for various numerical schemes solving a miscible displacement problem, introducing a new convergence result and proposing a correction for stability and accuracy across different conditions.

Contribution

It introduces a unified convergence framework for multiple numerical methods using the gradient discretisation approach, with a novel convergence result and a correction to improve stability and accuracy.

Findings

Centered convection discretisation performs poorly with variable viscosity and small diffusion.

Upstreaming schemes are not reliable for stability and accuracy.

The proposed correction improves scheme stability and accuracy across all test conditions.

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 $L^\infty(0,T; L^2(\Omega))$ 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 are 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.