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.