Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well-resolved. We design a second order Runge-Kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit, thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. Key words. Hyperbolic conservation laws with stiff relaxation, shock capturing difference method, Runge-Kutta methods, asymptotic limit AMS(MOS) subject classifications. 35L65, 35B40, 65M60 Typeset by A M S-T E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [42,36], in relaxing gas flow with thermal and chemical nonequilibrium [41,9], in kinetic theory of ra...

o TA a class while simultaneously taking it for credit. More importantly, for being an extremely valuable mentor, taking special care to introduce me to his colleagues. Further, it was he who provided the initial impetus for the work in Chapter V. 1 Whatever you do, do well. Even if you become a crook, just make sure you're a good one. 2 When one of my projects is going nowhere, I leave it (in the magic drawer) and work on a totally different project. When I return and start over, the answers "magically" jump out. iii Thanks to Professors Sichel, Van Leer and Powell for inviting me to Michigan. I have never regretted my decision -- hopefully, they have never done so either. My sincere gratitude to Professors Roe, Van Leer, Sichel, Powell and Harabetian, for serving on my committee, for reading through my dissertation at very short notice, and for their valuable insights and comments. Special thanks go, first, to Rosemary, who quickl

We explicitly construct a convex entropy function for a hyperbolic system with relaxation. This entropy is defined in the sense of Chen, Levermore and Liu [2], which resembles Boltzman's H-Theorem for kinetic equations. This construction follows the idea of Suliciu's energy function for a rate-type mixed hyperbolic-elliptic system [10]. Such an entropy will be useful in proving the entropy property of the relaxation schemes introduced by Jin and Xin for conservation laws [5]. Key words. Hyperbolic conservation laws with relaxation, entropy, H-Theorem. Research was supported in part by NSF grant No. DMS-9404157. Email address: jin@math.gatech.edu. Typeset by A M S-T E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [9,12], in relaxing gas flow with thermal and chemical nonequilibrium [3,11], in kinetic theory of rarefied gas dynamics [1], in viscoelasticity [8], multiphase and ...

A travelling waves analysis for a polyatomic gas with n internal energy storage modes is performed. In both the frozen flow and the equilibrium limit the existence and the qualitative behaviour of travelling waves are investigated. Methods from geometric singular perturbation theory are used to study these limits. Keywords: frozen flow, equilibrium, shock waves, travelling waves, singular perturbations 1 Introduction We consider a polyatomic gas with n different internal (quadratic) energy storage modes, e.g. vibrational and rotational modes. Many gases like hydrogen, oxygen and mixtures like air have such additional modes. It is known that relaxation mechanisms can affect the internal structure of shock waves and change their thickness [1, 2, 10]. To our knowledge no rigorous results on the existence of such waves have be proven. Our motivation to study this problem stems from the inreased interest in relaxation mechanisms in conservation laws and more specifically in gasdynamic equa...