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Un schéma aux volumes finis avec matrice signe pour les systèmes non homogènes

Abstract : This thesis is devoted to the analysis, the application and the two-dimensional extension, of a new finite volumes scheme (SRNH) proposed recently for a class of nonhomogeneous systems. The stability analysis of the scheme, first in the scalar case then in the case of systems of conservation laws, leads to a new formulation of the scheme which is based on the sign of the Jacobian matrix of the system under study. For Shallow Water equation with slope source term, one shows formally that the scheme SRNHS preserves the exact C-property introduced in the context of equilibrium schemes, by Bermùdez and Vázquez. The 1D and 2D numerical results, in particular in the case of a dam break over a step, show how much the scheme is really efficient. For two phase flows, nonhyperbolicity regions could appear, and the eigenvalues of the Jacobian matrix could become complex. It is shown that for weak nonhyperbolic configurations, one can calculate the sign of the Jacobian matrix using the algorithm of Newton-Schultz. For stiffer configurations, where the preceding method is no more adequate, one can use the method of density perturbation. In both cases, the numerical tests show that one approaches the exact solution of the Ransom problem with a high degree of accuracy, and that one preserves the stability of the calculations even on a grid of a relatively high degree of refinement.
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Contributor : Slah Sahmim <>
Submitted on : Tuesday, August 30, 2005 - 2:38:34 PM
Last modification on : Tuesday, May 5, 2020 - 1:03:20 PM
Long-term archiving on: : Friday, April 2, 2010 - 10:27:59 PM


  • HAL Id : tel-00010000, version 1


Slah Sahmim. Un schéma aux volumes finis avec matrice signe pour les systèmes non homogènes. Mathématiques [math]. Université Paris-Nord - Paris XIII, 2005. Français. ⟨tel-00010000⟩



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