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Numerical Modelling of Complex Dynamical Systems |
Numerical methods are used to investigate dynamical problems of
ever-increasing complexity. By
'dynamical problems' we understand time-dependent problems described by ordinary and/or partial
differential equations. This workshop
aims to provide an opportunity for interaction between scientists working on
some recent topics of intense research within this area. A great challenge confronting numerical analysis (as in other
fields) is how to effectively treat the interaction between dynamics on very
different spatio-temporal scales. In practice, one is
often most interested in the large or slow scale dynamics, and seeks to extract
the microscale influence on the macroscale;
yet a faithful model of the microscale should depend
on the macroscale as well. Unique to numerical
analysis is the incorporation of additional length scales associated with the discretization, such as the time stepsize
or spatial meshwidth which determine the resolution
of the method, and the interplay of these (e.g. numerical resonances). Current
work is aimed at stochastic models and appropriate methods; model reduction and
homogenization techniques, whereby small scale effects are incorporated through
appropriate averages; and equation-free methods, where the distinction between
model and simulation is blurred. All such methods demand new techniques for
their application and analysis. Another challenge is the development and analysis of numerical
methods that respect the dynamical properties (conservation laws, symmetries,
asymptotic behavior, etc.) of a problem being modelled
over long simulation times. Related to this is the development of effective
techniques for numerical modelling of problems in
which the dynamics is highly sensitive to perturbations, where rapid error
growth precludes accurate simulation of a particular solution trajectory over
long time, and where the goal of simulation is the computation of statistical
averages in an appropriate measure. For example, to distill accurate statistics
from simulations of molecular dynamics, it may be necessary for an integrator
to preserve volume and maintain constant energy. In other applications it is
essential for solutions to remain positive or to satisfy a maximum principle. The effective integration of partial differential equations
requires additional specialized techniques.
The presence of unbounded operators suggests the use of
semi-implicit or exponential integrators to deal with stiffness. The natural organizing structure of high
dimensionality suggests splitting methods.
These effects also drive error estimation and adaptivity.
Order reduction, finite domain effects, and the discord between conservation on one hand and the
nonlinear cascade to unbounded scales on the other, are important theoretical
aspects. [Back] |