Project Fact File

Title: Holistic discretisation of PDEs based upon modern dynamical systems theory
Category: maths
Area: computational and applied maths and engineering
No. of units: 1 or 2 or 4
Supervisor: Tony Roberts (Staff Profile)
Description:

The significant achievement is my development of techniques, based on centre manifold theory, for the rational and complete low-dimensional modelling of complicated dynamical systems. These techniques, which are born out of the recent explosion of interest in dynamical systems, apply to a variety of physical problems and lead to many new insights---some of which enable us to correct and complete classic approximations. Further, the systematic basis of the techniques allows us to devise new models in ways unimagined before. Recent applications are to the construction of discrete models, for numerical solution, using centre manifold techniques. The emerging advantages are: more robust numerical simulation due to the rational modelling of nonlinearity; a rational treatment of providing initial conditions; systematic derivation of boundary conditions for discretisations; systematic development to two-dimensional discretisations; and a completely novel theoretical justification of the numerical model. Further, it looks as if the approach can be extended to systematically modelling problems with convoluted subgrid scale structure. This project connects beautifully to the work of others on multiscale numerical methods---three invited speakers at extsc{iciam}~2003 spoke on multiscale topics. Amazingly the methodology extends to account for stochastic fluctuations in the physical problem. Our approach shows how nonlinearities give rise, in discretisations, to multiplicative noise and to essentially new noise processes when viewed on the time-scale of interest.

Student: anyone keen for 21st century applied mathematics.