Éditeur(s) : HAL CCSD
Résumé : International audience
In this paper, via the algebra of generalized functions, we investigate the generalized Riemann's problem associated to conservation laws with analytical coefficients. This allows us to transform the problem into a system of ordinary differential equations. In some particular cases, such that Burgers' and conservative Richard's equation, approximated solutions are obtained by the truncation of the so called Hugoniot-Maslov's chain and numerical simulations are also presented in the case of equations with polynomial coefficients.
Novi Sad Journal of Mathematics

hal-00771405
https://hal.archives-ouvertes.fr/hal-00771405

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
In this paper, we developed a theoretical study for nonconservative sytems in one dimension in order to construct numerical schemes for solving the Riemann problem. The nonconservative form of our model system required the use of a well-adapted theory in order to give us a sense of our problem. We chose a framework of generalized functions for solving a scalar hyperbolic equation with a discontinuous coefficient $sigma_t +usigma_x approx 0$, where u is the velocity solution of a Burgers's equation. After an explicit solution of the Riemann problem, we derived Godunov split schemes for computing an approximate solution of the Cauchy problem. We applied our study to a system modeling elasticity and a system modeling gas dynamics. Some stability properties of a scheme and its convergence to a generalized solution are proved for the first model. Numerical experiments confirmed this convergence result. For the second model, calculations of flows containing weak-to-moderate shocks showed that conservation errors are reduced when the mesh is refined but were not entirely eliminated.
ISSN: 0036-1429

hal-00770266
https://hal.univ-antilles.fr/hal-00770266
DOI : 10.1137/S0036142900378637

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
This article is intended as a preliminary report on the implementation of a finite volume multilevel scheme for the discretization of the incompressible Navier-Stokes equations. As is well known, the use of staggered grids (e.g., MAC grids [M. Perić, R. Kessler and G. Scheuerer, Comput. & Fluids 16 (1988), no. 4, 389-403; Zbl 0672.76018]) is a serious impediment for the implementation of multilevel schemes in the context of finite differences. This difficulty is circumvented here by the use of a colocated finite volume discretization [S. Faure, J Laminie and R. Temam, "Colocated finite volume schemes for fluid flows'', submitted; M. Perić, R. Kessler and G. Scheuerer, op. cit.], for which the algebra of multilevel methods is much simpler than in the context of MAC type finite differences. The general ideas and the numerical simulations are presented in this article in the simplified context of a two-dimensional Burgers equation; the two- and three-dimensional Navier-Stokes equations, which introduce new difficulties related to the incompressibility condition and the time discretization, will be considered elsewhere [see S. Faure, J Laminie and R. Temam, op. cit.; "Finite volume discretization and multilevel methods for the Navier-Stokes equations'', in preparation].
ISSN: 0885-7474

hal-00730193
https://hal.archives-ouvertes.fr/hal-00730193