| Date: Friday, January 28, 2011
Title: Numerical Approximation of Shock Waves in Non-Conservative Hyperbolic Systems
Abstract: Non-conservative hyperbolic systems arise in a wide range of applications, which makes their theoretical study and numerical approximation very important. The difficulty lies in the fact that shock relations depend not only on the immediate states ahead/behind the shock, but also on the viscous path that connects them. In 1995, the notion of path was introduced by Dal Maso, Le Floch and Murat to define generalized shock relations, and to develop a mathematical theory for weak solution of nonconservative hyperbolic systems. Around the same time, an alternative not unrelated theory was developed by Bianchini and Bressan based on the idea of vanishing viscosity solutions. While advances have been made on the theoretical front, those advances have been slow to translate into successful numerical methods. One attempt was the so-called path conservative numerical schemes proposed by Pares. How to choose a path and what is its influence on the computed solution are difficult questions. We shed light on these and other questions by considering an illuminating example using the Euler equations of gas dynamics in Lagrangian form, where the choice of linear path happens to give the correct jump conditions. This is joint work with Remi Abgrall, University of Bordeaux, France.
Speaker: Smadar Karni
Institution: University of Michigan
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