Abstract
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.
Highlights
Incompressible two-phase porous media flows are often modeled by the following set of equations
The energy stability of numerical methods appears to be a secondary point for a large part of the mathematical literature concerning porous media flows with capillary diffusion
Before addressing the two-phase flow problem (1)–(4), let us first focus on the simpler scalar problem (9)
Summary
Incompressible two-phase porous media flows are often modeled by the following set of equations. /@tsa þ $ Á va 1⁄4 0; ð1Þ where / denotes the porosity of the rock (supposed to be constant w.r.t. time), where sa denotes the saturation of the phase a and va denotes the filtration speed of the phase a. As depicted in [6] (see [7,8,9]), this problem can be reinterpreted as the generalized gradient flow [10] of the energy. This energy is made of the capillary energy, of the gravitational potential energy, and of a contribution related to the constraint (3): & 0.
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