This IMA Volume in Mathematics and its Applications MULTIDIMENSIONAL HYPERBOLIC PROBLEMS AND COMPUTATIONS is based on the proceedings of a workshop which was an integral part ofthe 1988-89 IMA program on NONLINEAR WAVES. We are grateful to the Scientific Commit­ tee: James Glimm, Daniel Joseph, Barbara Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for planning and implementing an exciting and stimulating year-long program. We especially thank the Work­ shop Organizers, Andrew Majda and James Glimm, for bringing together many of the major figures in a variety of research fields connected with multidimensional hyperbolic problems. A vner Friedman Willard Miller PREFACE A primary goal of the IMA workshop on Multidimensional Hyperbolic Problems and Computations from April 3-14, 1989 was to emphasize the interdisciplinary nature of contemporary research in this field involving the combination of ideas from the theory of nonlinear partial differential equations, asymptotic methods, numerical computation, and experiments. The twenty-six papers in this volume span a wide cross-section of this research including some papers on the kinetic theory of gases and vortex sheets for incompressible flow in addition to many papers on systems of hyperbolic conservation laws. This volume includes several papers on asymptotic methods such as nonlinear geometric optics, a number of articles applying numerical algorithms such as higher order Godunov methods and front­ tracking to physical problems along with comparison to experimental data, and also several interesting papers on the rigorous mathematical theory of shock waves.

Macroscopic limits of kinetic equations.- The essence of particle simulation of the Boltzmann equation.- The approximation of weak solutions to the 2-D Euler equations by vortex elements.- Limit behavior of approximate solutions to conservation laws.- Modeling two-phase flow of reactive granular materials.- Shocks associated with rotational modes.- Self-similar shock reflection in two space dimensions.- Nonlinear waves: overview and problems.- The growth and interaction of bubbles in Rayleigh-Taylor unstable interfaces.- Front tracking, oil reservoirs, engineering scale problems and mass conservation.- Collisionless solutions to the four velocity Broadwell equations.- Anomalous reflection of a shock wave at a fluid interface.- An application of connection matrix to magnetohydrodynamic shock profiles.- Convection of discontinuities in solutions of the Navier-Stokes equations for compressible flow.- Nonlinear geometrical optics.- Geometric theory of shock waves.- An introduction to front tracking.- One perspective on open problems in multi-dimensional conservation laws.- Stability of multi-dimensional weak shocks.- Nonlinear stability in non-newtonian flows.- A numerical study of shock wave refraction at a CO2/CH4 interface.- An introduction to weakly nonlinear geometrical optics.- Numerical study of initiation and propagation of one-dimensional detonations.- Richness and the classification of quasilinear hyperbolic systems.- A case of singularity formation in vortex sheet motion studied by a spectrally accurate method.- The Goursat-Riemann problem for plane waves in isotropic elastic solids with velocity boundary conditions.