Regular Article
Energy stable numerical methods for porous media flow type problems
INRIA, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000
Lille, France
* Corresponding author: clement.cances@inria.fr
Received:
28
February
2018
Accepted:
19
September
2018
Many problems arising in the context of multiphase porous media flows that take the form of degenerate parabolic equations have a dissipative structure, so that the energy of an isolated system is decreasing along time. In this paper, we discuss two approaches to tune a rather large family of numerical method in order to ensure a control on the energy at the discrete level as well. The first methodology is based on upwinding of the mobilities and leads to schemes that are unconditionally positivity preserving but only first order accurate in space. We present a second methodology which is based on the construction of local positive dissipation tensors. This allows to recover a second order accuracy w.r.t. space, but the preservation of the positivity is conditioned to some additional assumption on the nonlinearities. Both methods are based on an underlying numerical method for a linear anisotropic diffusion equation. We do not suppose that this building block is monotone.
© C. Cancès, published by IFP Energies nouvelles, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
