Ebook: Multidimensional Hyperbolic Problems and Computations

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.

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This volume is the proceedings of a two week workshop on multidimensional hyperbolic problems held during April 1989. The twenty-six papers in this volume emphasize the interdisciplinary nature of contemporary research in this field involving combinations of ideas from the theory of nonlinear partial differential equations, asymptotic methods, numerical computation and experiments. This volume includes several expository 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. In addition, there are two papers in the book devoted to open problems with this interdisciplinary emphasis. This book should be very interesting for any researcher pursuing modern developments in the theory and applications of hyperbolic conservation laws.

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This volume is the proceedings of a two week workshop on multidimensional hyperbolic problems held during April 1989. The twenty-six papers in this volume emphasize the interdisciplinary nature of contemporary research in this field involving combinations of ideas from the theory of nonlinear partial differential equations, asymptotic methods, numerical computation and experiments. This volume includes several expository 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. In addition, there are two papers in the book devoted to open problems with this interdisciplinary emphasis. This book should be very interesting for any researcher pursuing modern developments in the theory and applications of hyperbolic conservation laws.
Content:
Front Matter....Pages i-xiv
Macroscopic Limits of Kinetic Equations....Pages 1-12
The Essence of Particle Simulation of the Boltzmann Equation....Pages 13-22
The Approximation of Weak Solutions to the 2-D Euler Equations by Vortex Elements....Pages 23-37
Limit Behavior of Approximate Solutions to Conservation Laws....Pages 38-57
Modeling Two-Phase Flow of Reactive Granular Materials....Pages 58-67
Shocks Associated with Rotational Modes....Pages 68-69
Self-Similar Shock Reflection in Two Space Dimensions....Pages 70-88
Nonlinear Waves: Overview and Problems....Pages 89-106
The Growth and Interaction of Bubbles in Rayleigh-Taylor Unstable Interfaces....Pages 107-122
Front Tracking, Oil Reservoirs, Engineering Scale Problems and Mass Conservation....Pages 123-139
Collisionless Solutions to the Four Velocity Broadwell Equations....Pages 140-155
Anomalous Reflection of a Shock Wave at a Fluid Interface....Pages 156-168
An Application of Connection Matrix to Magnetohydrodynamic Shock Profiles....Pages 169-172
Convection of Discontinuities in Solutions of the Navier-Stokes Equations for Compressible Flow....Pages 173-178
Nonlinear Geometrical Optics....Pages 179-197
Geometric Theory of Shock Waves....Pages 198-202
An Introduction to front Tracking....Pages 203-216
One Perspective on Open Problems in Multi-Dimensional Conservation Laws....Pages 217-238
Stability of Multi-Dimensional Weak Shocks....Pages 239-250
Nonlinear Stability in Non-Newtonian Flows....Pages 251-260
A Numerical Study of Shock Wave Refraction at a CO2/CH4 Interface....Pages 261-280
An Introduction to Weakly Nonlinear Geometrical Optics....Pages 281-310
Numerical Study of Initiation and Propagation of One-Dimensional Detonations....Pages 311-314
Richness and the Classification of Quasilinear Hyperbolic Systems....Pages 315-333
A Case of Singularity Formation in Vortex Sheet Motion Studied by a Spectrally Accurate Method....Pages 334-366
The Goursat-Riemann Problem for Plane Waves in Isotropic Elastic Solids with Velocity Boundary Conditions....Pages 367-386

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This volume is the proceedings of a two week workshop on multidimensional hyperbolic problems held during April 1989. The twenty-six papers in this volume emphasize the interdisciplinary nature of contemporary research in this field involving combinations of ideas from the theory of nonlinear partial differential equations, asymptotic methods, numerical computation and experiments. This volume includes several expository 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. In addition, there are two papers in the book devoted to open problems with this interdisciplinary emphasis. This book should be very interesting for any researcher pursuing modern developments in the theory and applications of hyperbolic conservation laws.
Content:
Front Matter....Pages i-xiv
Macroscopic Limits of Kinetic Equations....Pages 1-12
The Essence of Particle Simulation of the Boltzmann Equation....Pages 13-22
The Approximation of Weak Solutions to the 2-D Euler Equations by Vortex Elements....Pages 23-37
Limit Behavior of Approximate Solutions to Conservation Laws....Pages 38-57
Modeling Two-Phase Flow of Reactive Granular Materials....Pages 58-67
Shocks Associated with Rotational Modes....Pages 68-69
Self-Similar Shock Reflection in Two Space Dimensions....Pages 70-88
Nonlinear Waves: Overview and Problems....Pages 89-106
The Growth and Interaction of Bubbles in Rayleigh-Taylor Unstable Interfaces....Pages 107-122
Front Tracking, Oil Reservoirs, Engineering Scale Problems and Mass Conservation....Pages 123-139
Collisionless Solutions to the Four Velocity Broadwell Equations....Pages 140-155
Anomalous Reflection of a Shock Wave at a Fluid Interface....Pages 156-168
An Application of Connection Matrix to Magnetohydrodynamic Shock Profiles....Pages 169-172
Convection of Discontinuities in Solutions of the Navier-Stokes Equations for Compressible Flow....Pages 173-178
Nonlinear Geometrical Optics....Pages 179-197
Geometric Theory of Shock Waves....Pages 198-202
An Introduction to front Tracking....Pages 203-216
One Perspective on Open Problems in Multi-Dimensional Conservation Laws....Pages 217-238
Stability of Multi-Dimensional Weak Shocks....Pages 239-250
Nonlinear Stability in Non-Newtonian Flows....Pages 251-260
A Numerical Study of Shock Wave Refraction at a CO2/CH4 Interface....Pages 261-280
An Introduction to Weakly Nonlinear Geometrical Optics....Pages 281-310
Numerical Study of Initiation and Propagation of One-Dimensional Detonations....Pages 311-314
Richness and the Classification of Quasilinear Hyperbolic Systems....Pages 315-333
A Case of Singularity Formation in Vortex Sheet Motion Studied by a Spectrally Accurate Method....Pages 334-366
The Goursat-Riemann Problem for Plane Waves in Isotropic Elastic Solids with Velocity Boundary Conditions....Pages 367-386
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