The study of Riemann problems has undergone a strong, steady growth in the last decade. The general direction of the research has headed toward understanding the wave structure of the solutions of more physically realistic systems. These systems fail either or both of the two main restrictions of the classical theory--that the system be strictly hyperbolic or genuinely nonlinear. The systems that have been studied tend to fall into the following broad classes: real gas dynamics (including combustion), visco-elastic materials, phase transitions, and multiphase flow in porous media. In addition to their usefulness in large-scale calculations, computational schemes have vastly improved the handling of discontinuity behavior.

This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Current Progress in Hyperbolic Systems: Riemann Problems and Computations, held at Bowdoin College in July 1988. The papers presented here provide a complete picture of recent research by some of the leaders in this field. Graduate students and beginning researchers will find this book a useful introduction to current work in this area.

Table of Contents

• D. Hoff and T.-P. Liu -- Shock wave solutions of the \(1d\) Navier-Stokes equations for compressible isentropic flow
• G. Schleiniger, M. C. Calderer, and L. P. Cook -- Embedded hyperbolic regions in a nonlinear model for visco-elastic flow
• I. Aavatsmark -- Capillary energy and the entropy condition for the Buckley-Leverett equation
• S. S. Antman and W. G. Szymczak -- Nonlinear elastoplastic waves
• M. Brio -- An example of a Riemann problem of second kind
• B. G. Bukiet -- Density profiles for diverging detonations
• J. Grove -- Anomalous waves in shock wave-fluid interface collisions
• D. S. Malkus, J. Nohel, and B. Plohr -- Time-dependent shear flow of a non-Newtonian fluid
• T. Zhang -- The Riemann problem for combustion
• E. Isaacson, D. Marchesin, and B. Plohr -- Transitional shock waves
• J. Trangenstein -- Three-phase flow with gravity
• B. Bohannon -- A system of conservation laws with a parabolic degeneracy
• J. Hunter -- Nonlinear surface waves
• B. L. Keyfitz -- A criterion for certain wave structures in systems that change type
• D. Marchesin and H. B. Medeiros -- A note on the stability of eigenvalue degeneracy in nonlinear conservation laws of multiphase flow
• R. Menikoff -- Analogies between Riemann problem for \(1-D\) fluid dynamics and \(2-D\) steady supersonic flow
• E. B. Pitman and D. G. Schaeffer -- Instability and ill-posedness in granular flow
• H. Gilquin and D. Serre -- Well-posedness of the Riemann problem; consistency of the Godunov's scheme
• V. Roytburd -- The Riemann problem for a system of conservation laws modeling phase transitions
• D. Wagner -- Detonation waves and deflagration waves in the one dimensional ZND model for high Mach number combustion
• G.-Q. Chen and A. Rustichini -- The Riemann solution to a system of conservation laws, with application to a non-zero sum game
• Z. Xin -- Asymptotic stability of planar rarefaction waves for scalar viscous conservation laws in several dimensions
• E.-Z. Fu, T. Tao, and Z.-H. Teng -- Riemann problem for a combustion model system: the existence and basic structure of the self-similar solutions
• R. Saxton -- Dynamic instability of the liquid crystal director
• H. Holden and L. Holden -- On the Riemann problem for a prototype of a mixed type conservation law. II