Work

Coupling techniques for nonlinear hyperbolic equations. I Self-similar diffusion for thin interfaces

Year: 2011

Type: article

Abstract: We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial-value problem, and these solutions need to be supplemented with further admissibility conditions. This paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns.

Source: Proceedings of the Royal Society of Edinburgh Section A Mathematics

DOI: 10.1017/s0308210510001459

Authors: Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch