Discrete shock pro les and their stability
نویسنده
چکیده
We discuss the existence and the asymptotic stability of shock prooles associated to nite diierence schemes. Assuming that the shock is stationnary, we look for steady solutions of the scheme. There are strong resemblances with traveling waves obtained by means of a parabolic approximation (the so-called viscous prooles). However, main diierences occur when the shock is under-compressive, these being the counterpart of the diierences between the dynamics of vector elds and the one of diieomorphisms. The stability is investigated by means of an Evans function, in the spirit of the recent analysis of viscous prooles by R. Gardner & K. Zumbrun 2]. However, the results diier qualitatively. Abstract Nous etudions ici les prools discrets de chocs stationnaires, pour des sch emas conservatifs aux dii erences nies. Nous d ecrivons les ressemblances (importantes) et les divergences (qui ne le sont pas moins) avec le cas des prools de viscosit e. La stabilit e de ces prools est analys ee au moyen d'une fonction d'Evans, avec le m^ eme l conducteur que dans le travail de R. Gardner et K. Zumbrun 2] pour les prools de viscosit e. Les conclusions sont cependant qualitativement dii erentes. Let @ t u + @ x f(u) = 0; x 2 R; t > 0; (1) be a system of conservation laws, where f : U ! R n is a vector eld, U being a convex open set in R n. We shall assume the strict hyperbolicity in the vicinity of distinguished points, u l and u r , but this property may fail elsewhere, with the consequence that under-compressive (as well as over-ones) shocks may occur.
منابع مشابه
On the stability of large amplitude semi-discrete shock pro les by means of an Evans function in in nite dimensions
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تاریخ انتشار 2007