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Finite volume methods on general meshes for nonlinear evolution systems

Abstract : In Chapter 1 we study a family of finite volume schemes for the numerical solution of degenerate parabolic convection-reaction-diffusion equations modeling contaminant transport in porous media. The discretization of possibly anisotropic and heterogeneous diffusion terms is based upon a family of numerical schemes, which include the hybrid finite volume scheme, the mimetic finite difference scheme and the mixed finite volume scheme. One discretizes the convection term by means of a family of schemes which makes use of the discrete unknowns associated to the mesh interfaces, and contains as special cases an upwind scheme and a centered scheme. The numerical schemes which we study are locally conservative and allow computations on general multi-dimensional meshes. We prove that the unique discrete solution converges to the unique weak solution of the continuous problem. We also investigate the solvability of the linearized problem obtained during Newton iterations. Finally we present a number of numerical results in space dimensions two and three using nonconforming adaptive meshes and show experimental orders of convergence for upwind and centered discretizations of the convection term.In Chapter 2 we propose a finite volume method on general meshes for the numerical simulation of an incompressible and immiscible two-phase flow in porous media. We consider the case that it can be written as a coupled system involving a degenerate parabolic convection-diffusion equation for the saturation together with a uniformly elliptic equation for the global pressure. The numerical scheme, which is implicit in time, allows computations in the case of a heterogeneous and anisotropic permeability tensor. The convective fluxes, which are non monotone with respect to the unknown saturation and discontinuous with respect to the space variables, are discretized by means of a special Godunov scheme. We prove the existence of a discrete solution which converges, along a subsequence, to a solution of the continuous problem. We present a number of numerical results in space dimension two, which confirm the efficiency of the numerical method.Chapter 3 is devoted to the study of a two-phase flow problem in the case that the capillary pressure curve is discontinuous with respect to the space variable. More precisely we assume that the porous medium is composed of two different rocks, so that the capillary pressure is discontinuous across the interface between the rocks. As a consequence the oil saturation and the global pressure are discontinuous across the interface with nonlinear transmission conditions. We discretize the problem by means of a numerical scheme which reduces to a standard finite volume scheme in each sub-domain and prove the convergence of a sequence of approximate solutions towards a weak solution of the continuous problem. The numerical tests show that the scheme can reproduce the oil trapping phenomenon.
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Submitted on : Thursday, December 1, 2011 - 6:20:41 PM
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Konstantin Brenner. Finite volume methods on general meshes for nonlinear evolution systems. General Mathematics [math.GM]. Université Paris Sud - Paris XI, 2011. English. ⟨NNT : 2011PA112244⟩. ⟨tel-00647336⟩



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