Stiff Sources and Numerical Methods for Conservation Laws

April 4 to April 8, 2005

at the

American Institute of Mathematics, Palo Alto, California

organized by

Doron Levy and Benoit Perthame

This workshop, sponsored by AIM and the NSF, will focus on unifying techniques which were developed in different application areas and exploring new application areas where existing techniques have not yet been applied.

Hyperbolic systems of conservation laws are being used for models in a variety of applications such as gas dynamics, acoustics, optics, and geophysics. It turns out that in many areas of interest, the dynamics is driven by balance laws that are conservation laws with source terms. Several important examples include:

  1. Shallow water equations in environmental sciences (flow of pollutants, global atmospheric simulations, oceanography, avalanches).
  2. Stochastic forces. Appear, e.g., in nano-devices and nano-bio-devices.
  3. Strong relaxation terms, such as strong absorption in Darcy law.
While numerical methods for conservation laws are well studied, the arising field of numerical methods for balance laws has been under-explored. The main difficulties in deriving stable and convergent numerical approximationsis due to the transition from the continuous framework to a discrete framework. It is a challenging task to obtain numerical schemes that are capable of both quantitatively and qualitatively capturing the behavior of the analytical solutions. Such complex behaviors include, e.g., steady states (that might contain non-smooth structures), and solutions that vanish in a nontrivial region (such as vacuum computations in gas dynamics or dry states in water flows). Recent major achievements in this field include stable numerical schemes that have been successfully used for predicting the flow of pollutants in rivers, for predicting snow avalanches (and the subsequent design of control systems that change the trajectory of the snow falling downhill), and for modeling of floods in open areas and in residential areas.

The area of numerical methods for balance laws has been very active in the past five years. Nevertheless, most of the activity in this area has been driven by specific applications. The motivations for this workshop are:

  1. Similar techniques were developed in different application areas.
  2. There are new application areas where existing techniques have not yet been applied.
It is the goal of this meeting to bring together researchers in these areas in order to classify and unify the current algorithms, to clearly identify the open problems in this field and to discuss future directions.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and working sessions.

Invited participants include F. Bouchut, S. Bryson, A. Chertock, B. Despres, F. Filbet, I. Gamba, A. Gelb, M. Gerritsen, J. Goodman, T. Hou, S. Jin, S. Karni, T. Katsaounis, R. Klein, D. Kroener, A. Kurganov, D. Levy, M. Lukacova, S. Noelle, B. Perthame, G. Russo, C. Simeoni, E. Tadmor, V. Zeitlin (Tseitline).

The deadline to apply for support to participate in this workshop has passed.

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