Advanced
Course on
Long Time Integrations
Barcelona, September
3-7,
2007
For many differential equations it is necessary to have methods
which are both accurate and fast. This need appears in long time
integrations (say, of the order of $10^9$ times some characteristic
time of the system) and also for moderate time intervals when massive
simulations are required for global and statistical purposes. Beyond
classical methods there is at present a strong interest in
``geometric'' methods which are intended to preserve some geometric
and/or structural properties of the system. It is worth to discuss the
suitability of different methods as a function of the problem at hand.
Furthermore moderate and long time integrations are used to detect
properties (regularity, chaoticity) of the system which can be done by
computing Lyapunov exponents and carrying out a frequency analysis.
Another purpose is the numerical computation of relevant objects such
as periodic and quasiperiodic solutions, etc, to study its stability,
bifurcations and continuation as a function of parameters. These
computations can require a very good accuracy. Finally, with methods
having a very small truncation error, the role of the propagation of
round off error should be studied, specially for regular (non-chaotic)
orbits.
The course will
take place at Room B1, located at
the ground level of the Court of Sciences (Pati de Ciencies), in
the right part of the historical building of the Universitat
de Barcelona (the address is at the end of this page). We will provide
wifi access for the participants.
How to arrive
Lecturers
G. Gómez (U. Barcelona), À. Jorba (U. Barcelona),
J. Laskar (IMCCE, Observatoire de Paris), B. Leimkuhler (Edinburgh U.),
J.M. Mondelo (U. Autònoma de Barcelona), C. Simó (U.
Barcelona).
Contents of the lectures
- J. Laskar
Symplectic and non-symplectic methods for long time integrations of
planetary systems. Frequency analysis of trajectories.
- B. Leimkuhler
Statistical mechanics from dynamics: Molecular dynamics algorithms,
computation of averages from trajectories, thermostats, error analysis
for averages.
Adaptive algorithms for molecular simulation: Nonequilibrium modelling,
quantum-mechanical/molecular mechanical models, parameterized molecular
dynamics.
- G. Gómez and
J.M. Mondelo
General methodologies: Continuation and multiple shooting methods.
Refined Fourier analysis procedures.
Applications: Periodic orbits. Invariant tori. Bifurcations. Invariant
manifolds. Homoclinic and heteroclinic connections.
- A. Jorba
A package for the automatic generation of Taylor integrators.
Extended arithmetics and applications.
- C. Simó
Principles of Taylor methods for analytic, non-stiff, ODE. Recurrent
computations. Selection of order and step-size. Round off errors.
Preservation of first integrals.
Some tests: simple Hamiltonian systems, three and N-body problems, etc.
Computation of Lyapunov exponents. The use in massive simulations.
Results and performance.
Slides and links
Carles Simó:No. 8 at http://www.maia.ub.es/dsg/2007/
J.M.Mondelo:Tuesday's lecture and Thursday's lecture
Àngel Jorba: Three sessions
Ben Leimkuhler:Six lectures, Local copy of the whole thing in a tar.gz file
Schedule
|
Monday
|
Tuesday
|
Wednesday
|
Thursday
|
Friday
|
09:00-10:00
|
Simó
|
Jorba
|
Leimkuhler
|
Leimkuhler
|
Gómez
|
10:00-11:00
|
Simó |
Simó
|
Leimkuhler
|
Leimkuhler
|
Jorba
|
11:00-11:30
|
Coffee
break
|
Coffee
break |
Coffee
break |
Coffee
break |
Coffee
break |
11:30-12:30
|
Leimkuhler
|
Laskar
|
Laskar
|
Laskar
|
Jorba
|
12:30-13:30
|
Leimkuhler
|
Laskar
|
Laskar
|
Laskar
|
Closing
|
13:30-15:30
|
|
|
|
|
|
15:30-16:30
|
Gómez
|
Mondelo
|
Jorba
|
Mondelo
|
|
16:30-17:00
|
Coffee
break |
Coffee
break |
Coffee
break |
Coffee
break |
|
17:00-18:00
|
Gómez
|
Mondelo
|
F. Borondo2
|
S. Blanes3
|
|
18:00-19:00
|
|
R. Barrio1
|
|
J. Palacián4
|
|
1 Title of the talk: Polynomial series integration method
and applications
2 Title of the talk: Homolclinic motions in quantum mechanics
3 Title of the talk: Composition and splitting methods
4 Title of the talk: Lunar theory: analytical vs numerical
approach
Preregistration
To attend you must preregister first. To do it, click
here.
As the attendance is
limited, the preregistration will be closed when the course is full.
Registration
The registration fee is of 50€ for researchers belonging to the
Consolider project i-MATH, and 100€ for other researchers. Details will
be e-mailed to preregistered participants.
Grants
There are some grants to help with the local expenses to young
students. If you are interested, mark the appropriate box when
preregistering, send your CV (in pdf format) to the address lti07
imub.ub.es, and ask a
senior researcher to e-mail a support letter to the same address,
including the phone of the supporter.
Note that finantial help will be granted with the following order of
priority: young researchers belonging to teams included in i-MATH,
other researchers belonging to these teams, other spanish
participants, non-spanish participants.
The deadline to ask for a
grant is July 13. Granted students will be
notified by e-mail on July 20.
Accomodation
Contact to: investigadorsresa.es, www.residencia-investigadors.es,
www.resa.es
Make sure that the reservation is done well in advance. Otherwise it
can be full.
Organizers
The organizers are Àngel Jorba and Carles Simó, with
the collaboration of Ariadna Farrés and Montse Navarro. For more
information, write them to the address lti07 at imub.ub.es (this
address is
not clickable, you should type it in your e-mail program; this has been
done to minimize the amount of spam we receive).
Support
The course is supported by i-MATH (Consolider project from the Spanish
Minsitry of Education and Science), IMUB and the research teams on
Dynamical Systems of the UB.
| IMUB |
| Institut de
Matemàtica |
| Gran Via de les Corts
Catalanes 585, |
| 08007 Barcelona, Spain |
|
|
| Phone: (+34) 93
402 13 85 |
| Fax: (+34)
93 403 59 63 |
|
Last modified:
Sept 10, 2007
