My research interests are generally in the areas encompassed by
the terms Scientific Computing, Numerical Analysis and Computational
Fluid Dynamics (CFD), although the main focus is on the development of new
numerical methods for approximating differential equations. In particular,
I am interested in approaches where the discrete form naturally inherits
properties of the underlying mathematical model (e.g. conservation, maximum
principles, scale invariance, upwinding, genuinely multidimensional physics,
equilibria) within as general and flexible a framework as possible (i.e.
unstructured meshes, arbitrary order of accuracy, a range of applications).
Below is a list of my main research interests, with links to more
detailed pages on some of them.
- Finite Volume Schemes:
- work with British Aerospace involving structured multiblock meshes
and the central difference, artificial viscosity approach of
Jameson, Schmidt and Turkel (AIAA--81--1259, 1981)).
- developed a general framework for constructing positive finite
volume scheme via slope reconstruction on unstructured meshes.
- Fluctuation Splitting Schemes:
- formed the basis of my Ph.D. thesis, which extended them to
the shallow water equations and combined them with a simple node
movement algorithm (all for steady state problems).
- constructed, with P.L.Roe, a generalised Flux Corrected Transport
algorithm for combining positivity and high order accuracy.
- currently working on high order, positive schemes for unsteady
problems and the possibility of constructing a version of the
approach which allows a discontinuous representation of the
underlying solution (analogous to the Discontinuous Galerkin
finite element scheme).
- Finite Element Schemes:
- their relationship with finite volume schemes and how this can
lead to the possibility of positive finite element schemes through
upwinding.
- Source terms:
- how they should be discretised so that they balance other terms
in the equations appropriately.
- Moving boundary problems:
- developing, with M.J.Baines and P.K.Jimack, a new moving finite
element method to model a whole range of problems involving moving
boundaries.
- Reaction-diffusion equations:
- on the one hand supplying numerical results to provide insight
in to the underlying asymptotics, while on the other using them
as a simple model of electromechanical activity of the heart in
an attempt to provide insight in to the onset of heart attacks.
- Adaptive mesh refinement:
- applied to coastal engineering and meteorological problems.
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