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Research

The numerical analysis group carries our research in a number of areas, primarily Differential Equations, Optimization and Approximation.

Differential Equations

In Differential Equations the emphasis is on research into nonlinear phenomena and the dynamics of numerical methods. For example, positive results about the conditions under which solutions of certain numerical methods converge to travelling waves of the underlying PDE have been given, together with negative results about conditions leading to spurious waves.

More recent work has focussed on wave propagation through irregular grids and on adaptive time-stepping mechanisms for partial differenctial equations.

Optimization

In optimization a major priority is the continuing development of effective production software for Linear and Quadratic Programming, taking account of factors such as degeneracy, round-off error and techniques for sparse matrix processing. These codes form the building blocks for large-scale nonlinear optimization codes being developed, and both theoretical and algorithmic developments are being researched. Aspects of nonsmooth optimization are studied in which the determination of a characteristic rank is a common factor. There are many applications of these techniques. The importance of Professor Fletcher's work was recently recognised by his election to FRS. See Professor Fletcher's home page for further information.

Approximation

Research into numerical methods for approximation problems is currently concerned with different aspects of fitting models to uncertain data, and the development and analysis of algorithms for problems in the analysis of measurements.

Scientific Computing

In scientific computing the emphasis is on numerical methods, analysis and simulation of partial differential equations arising from various applications. The research focuses mainly on constrained differential equations and incompressible Navier-Stokes equations, microstructure of liquid crystals, liquid crystal flows and complex fluids, and atomistic-to-continuum (multi-scale) coupling methods in material science. Other applications and computational topics such as gas bubbles rising in water, phase-field method for multi-phase flows, moisture transport in bread baking, PDE based image processing, splitting adaptive moving mesh for quenching and blow-up problems and airline optimal seat booking are of interest as well.