One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t +div(qf (u))−∆ϕ(u) = 0 by a piecewise constant function u D using a discretization D in space and time and a finite volume scheme. The convergence of u D to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on u D are used to prove the convergence, up to a subsequence, of u D to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of u D to u. Some numerical results on a model equation are shown.
Mathematics Subject Classification: 65M12
The nonlinear parabolic degenerate problemLet Ω be a bounded open subset of R d , (d = 1, 2 or 3) with boundary ∂Ω and let T ∈ R * + . One considers the following problem.The initial condition is formulated as follows:u(x, 0) = u 0 (x) for x ∈ Ω.(2) Correspondence to: R. Eymard
Abstract. Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete L 2 (0, T ; H 1 (Ω)) estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.Mathematics Subject Classification. 35K65, 76S05, 65M12.
International audienceThis paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions. It consists of two parts. The first part is devoted to the definition of an original notion of entropy solutions to the continuous problem, which can be adapted to define a notion of measure-valued solutions, or entropy process solutions. The uniqueness of such solutions is established. In the second part, the convergence of the finite volume method is proved. This result relies on (weak) estimates and on the theorem of uniqueness of the first part. It also entails the existence of a solution to the continuous problem
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