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About the SYNODE project
1. Objectives
We quote from the original SYNODE application
The scientific activity in the SYNODE project (Symmetries in Numerical
solution of Ordinary Differential Equations) includes the development
of general methodologies for taking advantage of special structures
and symmetries in the equations, with emphasis on structures which are
important in various application areas. Potential gains are
- improving the quality of the solution by producing numerical
solutions that possess symmetries and structures that are
important to the continuous equations.
- reducing computational cost by exploiting symmetries.
2. A brief history of the SYNODE activity
The idea behind the SYNODE project was born some time towards the end
of 1994 after discussions between Owren and Munthe-Kaas. Owren had
been involved in collaboration with people working in structural
dynamics, and was curious about the non-commutativity problems arising
in the study of rigid rotations. Munthe-Kaas on the other hand, had
been involved in the SOPHUS project in Bergen, where Lie theory had
been used together with tensor analysis to understand better how to
deal with elasticity
equations. Object oriented software also played an important part,
and it became important to apply similar abstractions in the numerical
analysis as those in the object oriented approach.
Munthe-Kaas understood already in the winter of 1995 how Runge-Kutta
theory could be generalized to commutative Lie groups, and summer
1995, an application to NFR, with proposal name SYNODE, was written
and turned out successful.
Support was given for the three years 1996-1998. As it often turns
out, it is difficult to make detailed plans about future research activity,
the SYNODE project was no exception. The main activity was on the
solution of ODEs on manifolds, still there was also some activity on
symmetry preservation and similar aspects. During a period of 2-3
years, the interest on the international level increased rapidly, and
the activity grew into an established field of research with
participants from many countries. By the end of 1998, close to 30
papers had been written, a meeting at Oppdal and several smaller
seminars. Arne Marthinsen, the first student in the project prepared
to defend his thesis.
3. The logo, what does it mean?
Originally, there were three partners in the SYNODE project,
from the universities of Trondheim, Bergen and Oslo.
The three circles in the equilateral triangle symbolize these three
universities and the edges allude to the collaboration.
The open ended lines perpendicular to these edges are to symbolize the
interaction with foreign partners.
4. The SYNODE II project
In the Spring 1998, it was decided that one should try to obtain an
extension of the support from the Norwegian Research
Council. Although, it was already clear that the activity had taken a
course somewhat different from what is indicated by the acronym
SYNODE, the project had gained some international recognition, and the
symbol was well known to everyone who was interested in the topic.
Thus, it was decided that the name and symbol should be kept, but that
we would try to alter the course somewhat in the direction of Partial
Differential Equations and to be more attentive to applications.
The internationally established name of the research field
is now Geometric Integration a notion intended to capture all
numerical integration methods which take special care to the
underlying geometric features of the continuous equations that we are
solving. The SYNODE group in Bergen has taken the consequence of this,
and now operates with this term on their web pages .
5. Where are we, where do we want to go?
The time of the SYNODE period will soon be over, but definitely, the
research activity will continue under a different name, and perhaps
the focus of the research will change slightly as the years go by.
If a single topic within the SYNODE activity should be singled out as
particularly important, it would be that of Lie group methods.
It is not entirely without pride and satisfaction, we now observe that
these methods are getting more and more popular abroad. From their
introduction when they were dismissed by many as ``Computationally
too expensive'' and ``of little practical value'' the
SYNODE group never stopped believing in their potential, and over the
5-6 years the project has been going on, several breaktrhoughs have
been made, substantially reducing the computational cost, and new
applications are found on a regular basis. The (now) recent meeting in
Beijing showed that researchers from China are using Lie group methods
successfully in obtaining conservative schemes for the KdV equation,
and several other invited speakers at that meeting propose to use Lie
group methods as a part of solving certain equations.
Looking ahead, one of the most important upcoming events is the
special year on Geometric integration to be held at the Center of
Advanced Studies in Oslo the academic year 2002-2003. Several well
known people have accepted to visit throughout that year.
At the end of that academic year, the important conference SciCADE is
to be held in Trondheim, and we believe it is a good opportunity to
make more publicity around this new field.
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