Anthony Michel

Publications (3)1.32 Total impact

• Source

  Available from: Raphaèle Herbin

  Article: Mathematical Study of a Petroleum-Engineering Scheme

  Robert Eymard, Raphaele Herbin, Anthony Michel
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  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 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. 1
  11/2002;
• Article: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

  Robert Eymard, Thierry Gallouït, Raphaèle Herbin, Anthony Michel
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  ABSTRACT: One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of 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 to{\it u}. Some on a model equation are shown.
  Numerische Mathematik 06/2002; 92(1):41-82. · 1.32 Impact Factor
• Article: Convergence of a Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations.

  Robert Eymard, Anthony Michel
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  ABSTRACT: One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t + div(qf(u)) '(u) = 0 by a piecewise constant function uD using a discretization D in space and time and a finite volume scheme. The convergence of uD to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on uD are used to prove the convergence, up to a subsequence, of uD 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 uD to u. Some numerical results on a model equation are shown.
  10/2000;