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My main interests are the numerical analysis of time
dependent partial differential and integral equations.
Some applications areas are outlined below.
Wave Propagation and Scattering - Time Domain PDEs
DEMO
Wave propagation and scattering problems occur in areas as diverse as
radar, underwater sonar, seismology, and ultrasonic probes in medicine.
One interesting area is in how to deal with a few obstacles
in an otherwise uniform medium (like a few objects flying in the air).
I think that using the overlapping grid idea may give good results here,
and I'm keen to investigate the numerical analysis and practical applications
of that idea.
Wave Propagation and Scattering - Time Domain Boundary Integral Equations
(with P J Davies, Strathclyde)
As an alternative to solving PDEs, one can instead solve time
dependent integral equations for quantities defined only on the 2D surface of
the scatterer. This removes the need to solve PDEs in a 3D spatial region
which can be an expensive process. However, the resulting retarded potential
integral equations (RPIEs) present a big challenge in understanding their
numerical analysis and in constructing efficient methods for their
approximation.
Reservoir Simulation
DEMO
One of the problems in simulating the behaviour of
oil and gas fields and groundwater flow in the presence of boreholes
(wells) and other defects is the huge difference in scale between
the borehole and the reservoir. This is particularly important in
welltest analysis, where the pressure at a wellbore is measured and
deductions are made about the properties of the reservoir
based on these measurements. One approach is to use
overlapping grids
to make simulation more convenient, and there is still much work to be
done in this area.
Convolution Integral Reaction Diffusion Equations (with G Lord)
Standard models of phase transitions and various biological processes
are described by reaction-diffusion equations. An alternative model based
on a convolution integral operator in place of the standard diffusion term
arises in some cases (like nerve axon firing) and presents some interesting
challenges in numerical and applied analysis. There is still much work to do
in producing accurate and efficient numerical methods for such problems,
and in proving that they work.