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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 connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equation are compared with the classical derivation of Hilbert and Chapman-Enskog. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution for the Boltzmann equation and the Leray-Hopf solution for the Navier-Stokes equation is considered.
Claude Bardos, François Golse, David Levermore

The Essence of Particle Simulation of the Boltzmann Equation

We describe the mathematical structure of recently developed family of consistent and convergent particle simulation methods for the Boltzmann equation.
H. Babovsky, R. Illner

The Approximation of Weak Solutions to the 2-D Euler Equations by Vortex Elements

It is shown that the Euler equations of two-dimensional incompressible flow, with initial vorticity in L p , p > 1, possess weak solutions which may be obtained as a limit of vortex “blobs”; i.e., the vorticity is approximated by a finite sum of cores of prescribed shape which are advected according to the corresponding velocity field. If the vorticity is instead a finite measure of bounded support, such approximations lead to a measure-valued solution of the Euler equations in the sense of DiPerna and Majda [7]. The analysis is closely related to that of [7].
J. Thomas Beale

Limit Behavior of Approximate Solutions to Conservation Laws

We are concerned with the limit behavior of approximate solutions to hyperbolic systems of conservation laws. Several mathematical compactness theories and their role are described. Some recent and ongoing developments are reviewed and analyzed.
Chen Gui-Qiang

Modeling Two-Phase Flow of Reactive Granular Materials

In this study, we examine a two-phase model proposed by Baer and Nunziato to describe the modes of combustion from deflagration to detonation in reactive granular materials. The model treats all phases in nonequilibrium and fully compressible. A compaction evolutionary equation, describing changes in volume fraction, provides model closure. In contrast to a pressure equilibrium model that has elliptic regions, the system of equations is hyperbolic except at points where the relative flow is locally sonic.
Pedro F. Embid, Melvin R. Baer

Shocks Associated with Rotational Modes

Many interesting systems of conservation laws in several space variables are isotropic in the sense of equivariance under rotations acting on the spatial components of the independent variable and on appropriate components of the state variable (e.g. the velocity vector in gas dynamics, or the velocity vector and the magnetic field vector in magnetohydrodynamics). As consequences of the isotropy, the conservation law which governs the propagation of plane waves in a given direction is not only independent of this direction, but also inherits a rotational symmetry, where now rotations act only on the transverse parts of the said components of the state variable. Generically — e.g. in magnetohydrodynamics, but not in gas dynamics — these transverse parts exhibit an interesting wave pattern including isolated linearly degenerate rotational modes. The talk is especially on the following three facts: These modes have not only contact discontinuities, but also shocks associated with them; for an arbitrarily given strictly stable and rotationally equivariant viscosity many of these shocks have viscous profiles; for an arbitrary choice of the four viscosity parameters in the usual dissipative version of magnetohydrodynamics there exist intermediate magnetohydrodynamic shocks which have a viscous profile.
Heinrich Freistühler

Self-Similar Shock Reflection in Two Space Dimensions

Self-similar oblique shock wave reflection for the equations of unsteady inviscid gas dynamics is among the outstanding unsolved problems in nonlinear PDE. This is so due not only to its’ standing as the simplest possible nontrivial shock wave interaction problem for these equations and the importance of the problem in engineering applications, but to the tantalizingly large amount of experimental and computational data which is available after half a century of intensive research since WW II.
Harland M. Glaz

Nonlinear Waves: Overview and Problems

The subject of nonlinear hyperbolic waves is surveyed, with an emphasis on the discussion of a number of open problems.
James Glimm

The Growth and Interaction of Bubbles in Rayleigh-Taylor Unstable Interfaces

The dynamic behavior of Rayleigh-Taylor unstable interfaces may be simplified in terms of dynamics of fundamental modes and the interaction between these modes. A dynamic equation is proposed to capture the dominant behavior of single bubbles and spikes in the linear, free fall and terminal velocity stages. The interaction between bubbles, characterized by the process of bubble merger, is studied by investigating the motion of the outer envelope of the bubbles. The front tracking method is used for simulating the motion of two compressible fluids of different density under the influence of gravity.
James Glimm, Xiao Lin Li, Ralph Menikoff, David H. Sharp, Qiang Zhang

Front Tracking, Oil Reservoirs, Engineering Scale Problems and Mass Conservation

A critical analysis is given of the mechanisms for mass conservation loss for the front tracking algorithm of the authors and co-workers in the context of two phase incompressible flow in porous media. We describe the resolution to some of the non-conservative aspects of the method, and suggest methods for dealing with the remainder.
James Glimm, Brent Lindquist, Qiang Zhang

Collisionless Solutions to the Four Velocity Broadwell Equations

In this note we shall examine special collisionless solutions to the four velocity Broadwell equations. These solutions are new and seem to have gone unnoticed by other investigators who have worked in this area1. These solutions are apparently stable; that is in numerical simulations they appear as the asymptotic state of the evolving system.
J. M. Greenberg, Cleve Moler

Anomalous Reflection of a Shock Wave at a Fluid Interface

Several wave patterns can be produced by the interaction of a shock wave with a fluid interface. We focus on the case when the shock passes from a medium of high to low acoustic impedance. Curvature of either the shock front or contact causes the flow to bifurcate from a locally self-similar quasi-stationary shock diffraction, to an unsteady anomalous reflection. This process is analogous to the transition from a regular to a Mach reflection when the reflected wave is a rarefaction instead of a shock. These bifurcations have been incorporated into a front tracking code that provides an accurate description of wave interactions. Numerical results for two illustrative cases are described; a planar shock passing over a bubble, and an expanding shock impacting a planar contact.
John W. Grove, Ralph Menikoff

An Application of Connection Matrix to Magnetohydrodynamic Shock Profiles

In this note we shall summarize how to use the connection matrix to show the existence of a viscous profile to the magnetohydrodynamic (MHD) equations. Although Freistuhler [1] has obtained various intermediate shock profiles in a different parameter regime, our technique is different and interesting to know. This is a technique based on algebraic topology and an extension of Con-ley’s index. As the detailed version will be appearing elsewhere [2], in this note we concentrate on how to use the connection matrix.
Harumi Hattori, Konstantin Mischaikow

Convection of Discontinuities in Solutions of the Navier-Stokes Equations for Compressible Flow

We report here on results concerning the existence, uniqueness, and continuous dependence on initial data of discontinuous solutions of the Navier-Stokes equations for one-dimensional compressible fluid flow:
$$ \left\{ \begin{gathered} {\upsilon_t} - {u_x} = 0 \hfill \\ {u_t} + p{(\upsilon, e)_x} = {\left( {\frac{{ \in {u_x}}}{\upsilon }} \right)_x} \hfill \\ {\left( {\frac{{{u^2}}}{2} + e} \right)_t} + {(up(\upsilon, e))_x} = {\left( {\frac{{ \in u{u_x} + \lambda T{{(\upsilon, e)}_x}}}{\upsilon }} \right)_x} \hfill \\ \end{gathered} \right. $$
with Cauchy data
$$ \left[ \begin{gathered} \upsilon \hfill \\ u \hfill \\ e \hfill \\ \end{gathered} \right](x,0) = \left[ \begin{gathered} {\upsilon_0} \hfill \\ {u_0} \hfill \\ {e_0} \hfill \\ \end{gathered} \right](x) $$
Here v, u, e, p, and T represent respectively the specific volume, velocity, specific internal energy, pressure, and temperature in a fluid; x is the Lagrangian coordinate, so that x = constant corresponds to a particle path; and and λ are fixed positive viscosity parameters.
David Hoff

Nonlinear Geometrical Optics

Using asymptotic methods, one can reduce complicated systems of equations to simpler model equations. The model equation for a single, genuinely nonlinear, hyperbolic wave is Burgers equation. Reducing the gas dynamics equations to a Burgers equation, leads to a theory of nonlinear geometrical acoustics. When diffractive effects are included, the model equation is the ZK or unsteady transonic small disturbance equation. We describe some properties of this equation, and use it to formulate asymptotic equations that describe the transition from regular to Mach reflection for weak shocks. Interacting hyperbolic waves are described by a system of Burgers or ZK equations coupled by integral terms. We use these equations to study the transverse stability of interacting sound waves in gas dynamics.
John K. Hunter

Geometric Theory of Shock Waves

Substantial progresses have been made in recent years on shock wave theory. The present article surveys the exact mathematical theory on the behavior of nonlinear hyperbolic waves and raises open problems.
Tai-Ping Liu

An Introduction to front Tracking

In fluid flows one can often identify surfaces that correspond to special features of the flow. Examples are boundaries between different phases of a fluid or between two different fluids, slip surfaces, and shock waves in compressible gas dynamics. These prominent features of fluid dynamics present formidable challenges to numerical simulations of their mathematical models. The essentially nonlinear nature of these waves calls for nonlinear methods. Here we present one such method which attempts to explicitly follow (track) the dynamic evolution of these waves (fronts). Most of this exposition will concentrate on one particular implementation of such a front tracking algorithm for two space, where the fronts are one-dimensional curves. This is the code associated with J. Glimm and many co-workers.
Christian Klingenberg, Bradley Plohr

One Perspective on Open Problems in Multi-Dimensional Conservation Laws

It is evident from the lectures at this meeting that the subject of systems of hyperbolic conservation laws is flourishing as one of the prototypical examples of the modern mode of applied mathematics. Research in this area often involves strong and close interdisciplinary interactions among diverse areas of applied mathematics including
Large (and small) scale computing
Asymptotic modelling
Qualitative modelling
Rigorous proofs for suitable prototype problems
combined with careful attention to experimental data when possible. In fact, the subject is developing at such a rapid rate that new predictions of phenomena through a combination of theory and computations can be made in regimes which are not readily accessible to experimentalists. Pioneering examples of this type of interaction can be found in the papers of Grove, Glaz, and Colella in this volume as well as the recent work of Woodward, Artola, and the author ([1], [2], [3], [4], [5], [6]). In this last work, novel mechanisms of nonlinear instability in supersonic vortex sheets have been documented and explained very recently through a sophisticated combination of numerical experiments and mathematical theory.
Andrew J. Majda

Stability of Multi-Dimensional Weak Shocks

In this paper we discuss the stability of weak shocks for a class of multi-dimensional systems of conservation laws, containing Euler’s equations of gas dynamics; we study the well-posedness of the linearized problem, and study the behaviour of the L 2 estimates when the strength of the shock approaches zero.
Guy Métivier

Nonlinear Stability in Non-Newtonian Flows

In this paper, we discuss recent results on the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid between parallel plates located at x = ±1, and driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and steady shear strain rate that results in steady states having, in general, discontinuities in the strain rate. We explain why every solution tends to a steady state as t → ∞, and we identify steady states that are stable; more details and proofs will be presented in [8].
J. A. Nohel, R. L. Pego, A. E. Tzavaras

A Numerical Study of Shock Wave Refraction at a CO2/CH4 Interface

This paper describes the numerical computation of a shock wave refracting at a gas interface. We study a plane shock in carbon dioxide striking a plane gas interface between the carbon dioxide and methane at angle of incidence α i . The primary focus here is the structure of the wave system as a function of the angle of incidence for a fixed (weak) incident shock strength. The computational results agree well with the shock polar theory for regular refraction including accurately predicting the transition between a reflected expansion and a reflected shock. They also yield a detailed picture of the transition from regular to irregular refraction and the development of a precursor wave system. In particular, the computations indicate that for the specific case studied the precursor shock weakens to become a band of compression waves as the angle of incidence increases in the irregular regime.
Elbridge Gerry Puckett

An Introduction to Weakly Nonlinear Geometrical Optics

Many natural phenomenae are governed by systems of nonlinear conservation laws that are — in a first approximation — hyperbolic. In this context, the understanding of the laws governing the propagation and interaction of small but finite amplitude high frequency waves in hyperbolic P.D.E.’s, and their interactions with “large scale” phenomenae (mean flows, shear layers, shock and detonation waves, etc.) is very important. Weakly Nonlinear Geometrical Optics (W.N.G.O.) is an asymptotic formal theory whose objective is precisely to do this.
In this paper we review the theory of W.N.G.O. as it stands currently. An important point is that W.N.G.O. is not (yet?) a complete theory. It fails at caustics, singular rays, etc. Since these are frequently regions of physical interest, a good theory for what happens there is important — and mostly nonexistent. The current status of these important open problems will also be (briefly) reviewed.
Rodolfo R. Rosales

Numerical Study of Initiation and Propagation of One-Dimensional Detonations

In this paper we briefly review some recent work centered on numerical simulation of initiation and propagation of reactive shock waves. This work is a joint project with A. Majda [1–3] with some very recent contributions by Majda’s Ph.D. student A. Bourlioux.
Victor Roytburd

Richness and the Classification of Quasilinear Hyperbolic Systems

Rich quasilinear hyperbolic systems are those which possess the largest possible set of entropies. Such systems have a property of global existence of weak solutions, whatever large is the bounded initial data. Although the full gas dynamics is not rich, many physically meaningful systems are. One gives below new examples and properties of the fully linearly degenerate case.
Denis Serre

A Case of Singularity Formation in Vortex Sheet Motion Studied by a Spectrally Accurate Method

Moore’s asymptotic analysis of vortex sheet motion predicts that the Kelvin-Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron, Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff-Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behavior of the spectrum, the calculations must be performed in very high precision. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the evolution of a large amplitude initial condition, the results indicate that Moore’s asymptotic analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs significantly away from that predicted by Moore. Convergence of the numerical solution beyond the singularity time is not observed.
M. J. Shelley

The Goursat-Riemann Problem for Plane Waves in Isotropic Elastic Solids with Velocity Boundary Conditions

The differential equations for plane waves in isotropic elastic solids are a 6 × 6 system of hyperbolic conservation laws. For the Goursat-Riemann problem in which the initial conditions are constant and the constant boundary conditions are prescribed in terms of stress, the wave curves in the stress space are uncoupled from the wave curves in the velocity space and the equations are equivalent to a 3 × 3 system. This is not possible when the boundary conditions are prescribed in terms of velocity. An additional complication is that, even though the system is linearly degenerate with respect to the c 2 wave speed, the c 2 wave curves cannot be decoupled from the c 1 and c 3 wave curves. Nevertheless, we show that many features and methodology of obtaining the solution remain essentially the same for the velocity boundary conditions. The c and c 3 wave curves are again plane polarized in the velocity space although the plane may not contain a coordinate axis of the velocity space. Likewise, the c 2 wave curves are circularly polarized but the center of the circle may not lie on a coordinate axis of the welocity space. Finally, we show that the c 2 wave curves can be treated separately of the c 1 and c 3 wave curves in constructing the solution to the Goursat-Riemann problem when the boundary conditions are prescribed in terms of velocity.
T. C. T. Ting, Tankin Wang
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