[SEMINARI] Solving Conservation Law and Balance Equations by Particle Management

[Enric Marco]

Title: Solving Conservation Law and Balance Equations by Particle 
Management

                                Dr. Yossi Farjoun

 Gregorio Millan Institute of Fluid Dynamics, Nanoscience and Industrial 
Mathematics

                              Universidad Carlos III de Madrid (Campus 
de Excelencia)

Lugar:  Seminario de Astronomia y Astrofisica, Edificio de Investigacion 
Jeroni Munyoz
Dia y Hora: Viernes 26 de Marzo de 2010 a las 12:00 horas

Invitado por la Unidad de Investigacion de Analisis Numerico (Prof. 
Antonio Marquina)
del Departamento de Matematica Aplicada


 Abstract: Conservation equations are at the heart of many interesting and
important problems. Examples come from physics, chemistry, biology, 
traffic and many more.
Analytically, hyperbolic equations have a beautiful structure due to
the existence of characteristics. These provide the possibility of
transforming a conservation PDE into a system of ODE and thus greatly
reducing the computational effort required to solve such problems.
However, even in one dimension, one encounters problems quickly.

The most obvious difficulty that needs to be dealt with has to do with
the creation of shocks, or in other words, the crossing of
characteristics. With a particle based method one would like to avoid
a situation when one particle overtakes a neighboring one. However,
since shocks are inherent to many hyperbolic equations and relevant to
the problems that one would like to solve, it would be good not to
``smooth away'' the shock but rather find a good representation of it
and a good solution for the offending particles.

In this talk we will discuss a new particle based method for solving
(one dimensional, scalar) conservation law equations. The guiding
principle of the method is the conservative property of the underlying
equation. The basic method is conservative, entropy decreasing,
variation diminishing and exact away from shocks. A recent extension
allows solving equations with a source term, and another provides
``exact'' solutions to the PDE. The method compares favorably to other
benchmark solvers, for example CLAWPACK, and requires less computation
power to reach the same resolution. A few examples will be shown to
illustrate the method, and some of its extensions.

Due to the current limitation to 1D scalar, the main application we
are looking at is traffic flow on a large network. Though we still
hope to manage to extend the method to either systems or higher
dimensions (each of these extensions has its own set of difficulties),
I would be happy to discuss further possible applications or
suggestions for extensions.



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/------------------------------------\
 Dr. Enric Marco
 Departament d'Astronomia i Astrofísica
 Facultat de Física
 Universitat de València
 C/  Dr. Moliner, 50
 46100 Burjassot, Spain
 Phone: (34)-96-354-3069
 Fax:(34)-96-354-3084
 enric.marco en uv.es
 http://www.uv.es/marco
 http://www.uv.es/aulaastronomia
 http://www.uv.es/daa
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