Schéma SRNHS Analyse et Application d'un schéma aux volumes finis dédié aux systèmes non homogènes

International audience This article is devoted to the analysis, and improvement of a finite volume scheme proposed recently for a class of non homogeneous systems. We consider those for which the corressponding Riemann problem admits a selfsimilar solution. Some important examples of such problems are Shallow Water problems with irregular topography and two phase flows. The stability analysis of the considered scheme, in the homogeneous scalar case, leads to a new formulation which has a naturel extension to non homogeneous systems. Comparative numerical experiments for Shallow Water equations with sourec term, and a two phase problem (Ransom faucet) are presented to validate the scheme. Cet article concerne l'analyse et l'application, d'un schéma proposé récemment por une classe de systèmes non homogènes. Nous considérons ceux pour lesquels le problème de Riemann correpondant admet une solution autosimilaire. Deux exemples importants de tels problèmes sont l'écoulement d'eau peu profonde au-dessus d'un fond non plat et les problèmes diphasiques. l'analyse de stabilité du schéma, dans le cas scalaire homogène, amène à une nouvelle écriture qui a une extension naturelle pour le cas non homogène. Des expériences numériques comparatives pour des équations de saint-Venant avec topographie variable, et un problème diphasique (Robinet de Ransom) sont présentés pour évaluer l'efficacité du schéma.

A selfsimilar solution to the focusing problem for the porous medium equation

In the focusing problem we seek a solution to the porous medium equation whose initial distribution is in the exterior of some compact set (e.g. a ball). At a finite time T the gas will reach all points of the initially empty region R. We construct a selfsimilar solution of the radially symmetric focusing problem. This solution is an example of a selfsimilar solution of the second kind, i.e. one in which the similarity variable cannot be determined a priori from dimensional considerations. Our solution also shows that in more than one space dimension, the velocity of the gas is infinite at the centre of R at the focusing time T.

Adiabatic Supernova Expansion into the Circumstellar Medium

We perform one dimensional numerical simulations with a Lagrangian hydrodynamics code of the adiabatic expansion of a supernova into the surrounding medium. The early expansion follows Chevalier’s analytic selfsimilar solution until the reverse shock reaches the ejecta core. We follow the expansion as it evolves towards the adiabatic blast wave phase. Some memory of the earlier phases of expansion is retained in the interior even when the outer regions expand as a blast wave. We find the results are sensitive to the initial configuration of the ejecta and to the placement of gridpoints.