2007 Clifford Lectures
Tulane University March 21st-24th, 2007

Talks

For abstracts click here

Clifford Lectures delivered by Eitan Tadmor, University of Maryland

Lecture 1 "On nonlinear entropy stability"

Lecture 2 "Critical thresholds in Eulerian dynamics"

Lecture 3 "Kinetic formulations and regularizing effects in quasilinear PDEs"

Lecture 4 "Aliasing, resolution and (in-)stabilities in convection dominated problems"

I will discuss the role of aliasing in high-resolution computations of linear and nonlinear convection-dominated problems. It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. We show that due to weighted L2-stability, the N-degree Fourier solution is algebraically stable (in the sense that its L2 amplification does not exceed O(N)). Yet, the Fourier method is weakly L2 unstable, in the sense that it does experience such O(N)-amplification. The exact mechanism of this weak instability is due the aliasing phenomenon.

I will also consider the nonlinear regime, where de-aliasing is often the method of choice to maintain the balance of quadratic energy. We will prove that de-aliasing is responsible for spurious oscillations. Three practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their O(N) amplification. Happily, with enough resolution nothing worse can happen. And finally, high-order diffusion, e.g., through smoothing of spectral viscosity, is essential for nonlinear stability.

Talks by invited speakers

Alberto Bressan, Penn State University

"Convergence rates of vanishing viscosity approximations"

Pierre Degond, Université Paul Sabatier Toulouse 3

"Fluid models with localized kinetic upscaling"

Manoussos Grillakis, University of Maryland

"On the motion of a surface in four dimensional space"

Shi Jin, University of Wisconsin

"Hamiltonian systems and Liouville equations with discontinous Hamiltonians: Computation of high frequency waves in heterogeneous media"

David Levermore, University of Maryland

"Weakly nonlinear-dissipative approximations of hyperbolic-parabolic systems with entropy"

Hyperbolic-parabolic systems have spatially homogenous equilibria. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern pertrubations of these equilibria. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natrual Hilbert space. We show that under a mild structural hypothesis, this approximation has global weak solutions for all initial data in that Hilbert space. This theory applies to the compressible Navier-Stokes system. The resulting approximate system is an incompressible Navier-Stokes system coupled to equations that govern the acoustic modes. The solution of this approximate system is unique if the incompressible modes are uniquely determined.

Doron Levy, Stanford University

"Group dynamics of phototaxis"

Microbes live in fluctuating environments that are often limiting for growth. They have evolved several sophisticated mechanisms to sense changes in important environmental parameters such as light and nutrients. Most bacteria also have complex appendages that allow them to move, so they can swim or crawl into optimal conditions. This combination of sensing changes in the immediate environment and transducing these changes to the motion organisms, allows for movement in a particular direction: a phenomenon known as ''chemotaxis'' or ''phototaxis''.

Using time-lapse video microscopy we have monitored the movement of Cyanobacteria (which are phototaxis, i.e., bacteria that move towards light). These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. The various patterns of movement that we observe appear to be a complex function of cell density, surface properties and genotype. Very little is known about the interactions between these parameters.

In this talk, we will present a hierarchy of new models for describing the motion of phototaxis, that were constructed based on the experimental observations. The first model is a stochastic model that describes the locations of bacteria, the group dynamics, and the interaction between the bacteria and the medium in which it resides. The second model is a new multi-particle system that is obtained from a discretization of the first model. Our third model is obtained as the continuum limit of the second model, and as such it is a system of nonlinear PDEs. Our main theorems clarify the sense in which the system of PDEs can be considered as the limit dynamics of the multi-particle system. We conclude with several numerical simulations that demonstrate the properties of our models. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute) and Tiago Requeijo (Math, Stanford).

Hailiang Liu, Iowa State University

"An alternating evolution approximation to hyperbolic conservation law"

In this talk I shall present a novel alternating evolution (AE) approximation to hyperbolic conservation laws in arbitrary spatial dimension. We prove the convergence of the approximate solutions towards an entropy solution of scalar multi-D conservation laws. It is also shown that such an approximation is extremely accurate in the sense that if initial data is prepared, then no method error is induced as time evolves, and the exact entropy solution is precisely captured. These features render such an approximation ideal to be used for design of high resolution numerical schemes for solving hyperbolic conservation laws. The usual obstacles caused by jumps crossing computational cell interfaces are not felt when two variables in approximation are sampled alternatively, and reconstructed independently. We also discuss the designing principle for constructing AE schemes, with illustration of some preliminary ones, for which stability properties are established.

Chi-Wang Shu, Brown University

"Discontinuous Galerkin finite element method: survey and recent development"

Hongkai Zhao, University of California, Irvine

"Fast sweeping method for static convex Hamilton-Jacobi equation"

Contact Alexander Kurganov kurganov@math.tulane.edu for further information.

Mathematics Department Tulane University 6823 St. Charles Ave New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063

Last Updated: April 1, 2007