The solution of challenging problems modelling knock in a car engine and atmospheric pollution using an adaptive space-time error control code will be considered. In particular a comparison will be made with more traditional numerical techniques used in computational fluid dynamics. The new method used is based on a scheme which uses flux limiter techniques, approximate Riemann solvers and operator splitting to produce high quality solutions to complex flow problems. The issue of how to integrate in time when using such a scheme involves considerations such as positivity and what time accuracy/timestep to use. The talk will emphasise the need to control both the spatial and temporal errors in a consistent way and will show that adaptive methods allow the possibility of obtaining greater insight into the physical problem.
The Internet offers many useful services for mathematicians and others, and opens up new ways to publish and to retrieve information. In this talk we give a brief survey of some of the tools and information of interest to mathematicians available on the Internet.
Many of the World Wide Web pages shown in the talk can be found directly
Many situations in fluid dynamics are posed on unbounded domains and, when solving such problems numerically, it therefore becomes necessary to truncate the domain. Some boundary condition must be devised for the artifical (outflow) boundary that will not seriously affect the solution in the interior.
A new form of boundary condition for this purpose was introduced by Papanastasiou and colleagues (1992) specifically for finite element methods. We shall describe its properties in the context of convection dominated diffusion problems. The boundary condition is unusual because, in the weak formulation, it appears to impose no boundary condition at all at the outflow, thus making the problem ill-posed.
A short but elementary introduction is first given to neural networks, with an emphasis on feed-forward networks and their link to approximation theory and algorithms. Specific models that correspond to ridge function and radial basis function approximations are then detailed, and a variety of useful applications are briefly discussed - including image recognition, flood estimation, and control engineering. The problems that arise often involve "unconventional" types of data - not well covered in the approximation literature - such as integer data, high-dimensional data, very little data, and data which is computed from previous data. Two relatively novel techniques using Gaussian radial basis functions - separation of variables and orthogonalisation - are also briefly described.
with Bernd Fischer (Medical University of Luebeck),
David Silvester (UMIST) and Andy Wathen (University of Bristol)
Iterative methods of Krylov subspace type often provide attractive solution techniques for large sparse linear systems, particularly when used in conjunction with appropriate preconditioners. This talk will describe the applicability and convergence of preconditioned minimum residual methods for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Some theoretical and numerical results will be presented, mainly concerning the comparison of indefinite versus positive definite preconditioning. Both methods considered can be shown to have a close relationship with solving a system of normal equations.
A PostScript copy of the associated Strathclyde Mathematics Research Report "Minimum Residual Methods for Augmented Systems" (no. 15, June 1995) is available by clicking here or via anonymous ftp from ftp.strath.ac.uk (Maths/Staff/rep95_no15.ps).