arXiv subjects

Kyle S. Talbot

Publications and source records attributed to Kyle S. Talbot.

5 recordsLinked to original sources

arXiv2017

Unified convergence analysis of numerical schemes for a miscible displacement problem

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.

math.NA↗

arXiv2017

Analysis of miscible displacement through porous media with vanishing molecular diffusion and singular wells

This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium, with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with $L^2$ wells [\emph{Z. Anal. Anwendungen}, 23(2):335--351, 2004]. We give a novel treatment of the diffusion-dispersion term, which requires delicate use of the Aubin--Simon lemma to ensure the strong convergence of the pressure gradient, owing to the troublesome lower-order terms introduced by the localisation procedure.

math.AP↗

arXiv2016

Convergence in $C(\lbrack0,T\rbrack;L^2(Ω))$ of weak solutions to perturbed doubly degenerate parabolic equations

We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic $p$-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or additional regularity of the solution. However, when uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to perturbations of the data. Beginning with a proof of temporally-uniform, spatially-weak convergence, we strengthen the latter by relating the unknown to an underlying convex structure that emerges naturally from energy estimates on the solution. The double degeneracy --- shown to be equivalent to a maximal monotone operator framework --- is handled with techniques inspired by a classical monotonicity argument and a simple variant of the compensated compactness phenomenon.

math.AP↗

arXiv2015

Uniform temporal convergence of numerical schemes for incompressible miscible displacement

The Hybrid Mimetic Mixed (HMM) family of discretisations includes the Hybrid Finite Volume method, the Mimetic Finite Difference method and the Mixed Finite Volume method. This note demonstrates that HMM discretisations of the equations describing the single-phase, miscible displacement through a porous medium of one incompressible fluid by another converge uniformly in time for the concentration variable.

math.NA↗

arXiv2014

On a miscible displacement model in porous media flow with measure data

We establish the existence of a solution to a non-linearly coupled elliptic-parabolic system of PDEs describing the single-phase, miscible displacement of one incompressible fluid by another in a porous medium. We consider a velocity-dependent diffusion-dispersion tensor and model the action of injection and production wells by measures on the domain. We obtain the solution by passing to the limit on problems with regularised well data.

math.AP↗