HYperbolic and Kinetic Equations :
Asymptotics, Numerics, Analysis
of the




HYKE numerical gallery


Short presentation of the team E1

Sites of E1 Valencia + Bilbao, Madrid (Complutense, Politec.)
URL http://gata.matapl.uv.es/hyke/
Team Organizer R. Donat

The composition of the E1-team includes scientists from several spanish universities all of whom have been actively involved for at least one decade in research projects related to hyperbolic and kinetic equations. All members of the team are involved in PhD programs in Applied Mathematics in their respective universities. Training includes post-doctoral fellows, which have visited our host universities under several EC-TMR programs. In addition, various members of the team have also participated as lecturers in several summer-schools on topics related to their fields of expertise.

The research activity of the UV-branch has been mainly devoted to numerical aspects of high resolution simulations for systems of conservation laws (task 15 ). In particular, the team has developed a numerical technique (Marquina's scheme) that has proven to be fairly robust in a variety of well known (numerically) delicate situations. There is a long term collaboration with the astrophysics research group (at the UV, with ramifications to Germany and Italy) that has lead to fruitful numerical simulations of astrophysical flows (task 16 ). In recent times, research effort has been devoted to lowering the cost of some high resolution schemes for conservation laws, where the use of multi-scale techniques has led to significant improvements. It is expected that the combination of these techniques with an efficient parallel implementation of the algorithms will provide the team with the necessary ingredients to attempt the numerical simulation of complex flows (like multi-material compressible flows in 2 and 3D).

Both the UCM and the UPV branches have a wide expertise on the theoretical aspects of nonlinear PDE's and Control Theory. These two branches of the team have kept a close connection over the years that has lead to many fruitful collaborations. Research efforts at the UPV have been devoted to study the well posedness of the Cauchy problem for dispersive equations, viscous conservation laws, kinetic equations for quantum particles and coagulation and fragmentation models (task 5 ), as well as the qualitative study of their long time behavior, including singularity formation (task 3 ). In recent times, the UCM-branch has been involved in the analysis of singular behavior of solutions kinetic models arising in plasma physics and in classical `aggregation-fragmentation' models (task 5 ). Wave propagation phenomena in highly heterogeneous media have also been analyzed using homogenization and asymptotic methods techniques. The group is presently involved, among other things, in the asymptotic and numerical analysis of hyperbolic problems arising in the dynamics of dislocations and semiconductor dynamics (task 2 ).

The team has well established research links within the network, in particular with the following network teams: team E2 (semiconductor dynamics), team F2 (fluid-solid interaction and phase transition models), F1 (quantum evolution PDEs, diffusive limits), S1 (coagulation-diffusion models).

List of key scientific staff:

- R. Donat (TO) (U. Valencia 30%);  - F. Arandiga (UV, 15%);  - V. Candela (UV, 15%);  - P. Mulet (UV, 30%);  - A. Marquina (SAB) (UV, 30%);  - M. Escobedo (TC) (U. Pais Vasco, 30%);  - L. Vega (UPV, 30%);  - J. Aguirre (UPV, 20%);  - A. Carpio (U. Complutense, Madrid, 25%);  - M. A. Herrero (UCM, 25%);  - J. J. Velazquez (UCM, 25%);  - E. Zuazua (UCM, 30%);  - C. Castro (U. Politecnica Madrid, 30%).

Significant Publications:

[1] R. Donat, J.A. Font, J.M. Ibanez, A. Marquina, A Flux Split Algorithm Applied to relativistic Flows , Jour. Comp. Phys. 146 (1998), 58--81.

[2] M. Escobedo, M.A. Herrero, J.J.L. Velazquez, A nonlinear Fokker-Planck equation modeling the approach to thermal equilibrium in a homogeneous plasma , Transactions of AMS, 350 (10) (1998), 3837--3901.

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