#
Stiff Sources and Numerical Methods for Conservation Laws

April 4 to April 8, 2005
at the

American Institute of Mathematics,
San Jose, California

organized by

Doron Levy and Benoit Perthame

## Original Announcement

This workshop 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:

- Shallow water equations in environmental sciences (flow of pollutants,
global atmospheric simulations, oceanography, avalanches).
- Stochastic forces. Appear, e.g., in nano-devices and nano-bio-devices.
- 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:

- Similar techniques were developed in different application areas.
- 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.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.