main_pic02.gif (34316 bytes)    Conference in Honor of
James Greenberg's
60th Birthday


Liapunov Functions and PDEs

Marshall Slemrod

As is well known Liapunov functions can used to prove stability of equilibria for a given dynamical system. However given a dynamical system with some parameters ( e.g. controls ) which may be chosen by the mathematician or engineer, Liapunov's second method can be used as a design mechanism. This talk sketches a history of this idea and how it has been recently used in finding approximate solutions to the Boltzmann equation.

Marshall Slemrod
Mathematics Dept.
Univ. of Wisconsin
Madison WI 53706

Chaos and Fluid Mixing

James Glimm

Multiscale problems define the cliffs to be scaled for the science of our times. Chaotic fluid mixing, as a special case, illustrates the general features of such problems. Mixing and multiphase flow are dominant phenomena for flow in porous media, pipeline transport of mixtures, atmospheric climate models, and inertial confinement fusion, among other topics.

Basic methods to study chaotic fluid mixing reach across the length and time scales; these methods include analytic studies, direct numerical simulation, moment expansions, averaged equations and closure, and simple phenomenological models.

In this presentation, we will develop a general approach and set of methods for the solution of multiphase problems. These methods will be illustrated through recent work of the suthor and collaborators. High resolution simulation methods (Front Tracking) prevent diffusion across an interface and allow detailed late time direct numerical simulation in agreement with experiment. But even visions of teraflops will not
suffice, and progress with modeling or averaged equations will also be discussed. As usual, closure, physical consistency, and validation are the major issues for modeled equations.

James Glimm
Department of Applied Mathematics and Statistics
SUNY at Stony Brook, Stony Brook, NY 11794


John Ockendon

Slamming is the phenomenon of high velocity impact between a solid and a nearly inviscid liquid. To lowest order, gravity, viscosity and even compressibility are ignored. This talk will review some of the formulations that have been proposed for various slamming configurations, and recent progress that has been made to unify theories for slamming and skimming.

John Ockendon
Oxford Centre for Industrial and Applied Mathematics
Oxford University, UK

Toward Field Theories for Bodies Undergoing Disarrangements

David R. Owen

The physical behavior at multiple length scales of a variety of technologically important systems has generated challenging problems both in the formulation and in the analysis of appropriate mathematical models. Progress in these problems often has required a sharpening of focus through technical refinements and numerical implementations of already existing and newly emerging models. In this talk I will discuss the issue of how one might broaden the field theories of continuum mechanics in order to incorporate systematically the process of modeling at multiple length scales.

David R. Owen
Department of Mathematical Sciences, Carnegie Mellon

The approximation of the spectra of non-compact operators arising in buckling analysis

Manil Suri

A linearized model for buckling and stress-stiffening has been recently implemented in the hp code STRESSCHECK. This model does buckling analysis for the fully three-dimensional problem at hand, rather than some asymptotic (dimensionally reduced) limit. It finds the smallest positive multiple qmin of an existing (pre-buckling) stress state s0 that will result in buckling. The use of the hp finite element method enables solutions over singular domains to be well approximated, and ensures that no locking takes place even when the domain is very thin.

However, a potentially serious danger of the method is that it characterizes qmin as the lowest positive spectral value of a non-compact operator T. Such non-compact spectral value problems can be notoriously ill-behaved, due to the presence of spurious approximate eigenvalues, which can completely pollute the results.

We establish that if the domain is thin enough (in a sense made precise), one can recover the eigenvalues of interest provided the stress s0 is bounded. Moreover, for problems of interest (such as domains with corners and edges) where s0 may be unbounded, we show that one can still approximate the buckling eigenvalues of interest, even though the problem may not be well-posed.

Manil Suri
Department of Mathematics and Statistics
University of Maryland, Baltimore County

Hyperbolic Models of Traffic Flow

Michel Rascle

I will talk on a clsaa of Continuum Models of Traffic Flow, that I have introduced a few years go with a student : A. Aw. The basic question is : does a driver respond to a {\em space} or to a (Lagrangian) {\em time} gradient of the density ? We showed then that the first answer (the gas dynamics system) is completely wrong, and should be replaced by the second one, i.e. our model. I will then describe some mathematical questions (e.g. vacuum ...) and consider extensions of this model, e.g. zero-relaxation limit.
Finally, I will talk about a direct derivation from microscopic equations, joint work with A. Klar et al. This recent work uses exactly the same Lagrangian ideas developped independently by Jim Greenberg, but the motivation was different.

Michel Rascle

Navier Stokes Flow and Laws at Interfaces and Rough Boundaries

Willi Jäger

In this lecture a survey is given covering the results of asymptotic analysis for problems arising in flow along a rapidly oscillatory surface or in a partially porous medium. The transmission laws connecting the free flow and the filtrations flow in a porous media will be discussed, effective boundary conditions on an approximating ``smooth" boundary surface replacing a rough one will be derived. The dependence of the drag force on the scale of roughness is analysed in the nonturbulent situation. The effective terms and quantities can be numerically computed, errors of the approximations are estimated.

The results are in agreement with experimental measurements. The report is dealing with results obtained by Mikelic, N.Neuss and Jäger.

Prof. Dr. Dr. h.c. Willi Jäger
Applied Analysis
Institut für Angewandte Mathematik
Im Neuenheimer Feld 294
D-69120 Heidelberg

Source Terms for some Hyperbolic Problems

Alain-Yves LeRoux, Marie-Nöelle LeRoux

This topic corresponds to a good and creative cooperation with our friend Jim Greenberg, who visited us several times in Bordeaux.

The results obtained for the scalar equation, for which the proofs of convergence can be proposed, show that the new ideas allow to control and stabilize numerically the steady states, and also to reduce the CPU time since a larger time step may be used in many cases in practice, better than for other numerical schemes.

We propose to review two families of schemes: the Well Balanced (WB) schemes developped with Jim Greenberg, and the Stationary Profiles (SP) schemes developped more recently.

In a WB scheme, the source term is concentrated on the edges of the cell and interferes with the flux in the so called Riemann solver, which controls the exchanges thru this interface. Such a technique allows to use a constant "cellwise" approximate solution and gives an easy projection step in a Godunov method.

In a SP scheme, the source term is not modified and the profile of the solution in the step is shaped as a steady solution balancing the flux term and the source term. The projection step becomes however a little harder to manage, but the precision to upper order is easier to perform, by keeping the stability.

Next these two families of schemes are adapted to 2x2 hyperbolic systems, and one get this way some very stableschemes well adapted to the simulation of mathematical models involving equilibria as for environmental applications (water flooding, rivers, meteorology,...) or other applications such as a flow in a non uniform tube, in biomechanics, etc.

Error Estimation for Strong Shock Hydrodynamics

John W. Grove

The analysis and modeling of error production in strong shock hydrodynamics using statistical methods is discussed in the context of a set of simple one-dimensional test problems. Assuming known distributions for a set of initialization and flow parameters we compare the
results from simulations using both shock capturing and front tracking with a fine grid fiducial solution and study the probability distribution of the error residual. The goal of this research is to gain insight into the development of probability models for error generation and propagation in hydrodynamic flows with the ultimate goal of obtaining quantitative predictive models for the distribution of solution errors in a numerical simulation.

John W. Grove
Methods for Advanced Scientific Simulations Group
Computer and Computational Science Division
Los Alamos National Laboratory

Conference Organizers:
Irene Fonseca, Morton Gurtin, Stuart Hastings, David Kinderlehrer, David Owen, and William Williams


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