Exponential Integrators (Applied Nonlinear Dynamics)

A common procedure for the numerical solution of partial differential equations is the method of lines, which results in a large system of ordinary differential equations. These ordinary differential equations are often stiff, and thus implicit methods are commonly used to solve them.

However, implicit methods are costly because they solve a system of (algebraic) equations at every step. Exponential integrators provide an alternative. They are constructed using the (matrix) exponential; hence the name. The matrix exponential is the exact solution of a linear equation and thus a good building block for numerical methods. Possible topics for research include a theoretical analysis of the method, efficient algorithms for computing the exponential of both small and large matrices, and applying the method on large-scale practical problems, for example the simulation and analysis of processes on complex networks or the pricing of financial instruments.

Nonlinear dynamics and its applications at Leeds has for many years enjoyed reputation for a distinctive interdisciplinary approach. The Centre for Nonlinear Studies was established at Leeds in 1984 to enhance existing and foster new research collaborations between mathematicians, scientists and engineers throughout the university and beyond. Twenty five years later, the research group retains its character as an applications driven centre, applying dynamical systems theory to a range of natural phenomena. It has recently expanded with the appointment of several new members of staff, bringing the total to ten permanent members of staff working with five postdocs and postgraduate students.

Applied Nonlinear Dynamics is a vibrant research area lying at the heart of problems of fundamental and practical importance. It employs a wealth of mathematical techniques, from statistical to geometrical, from computational to algebraic, and from qualitative to analytical. The main concern is systems that change with time, where the presence of nonlinearities can produce hugely complicated behaviour. The range of activities in Applied Nonlinear Dynamics is extremely broad. Core areas of investigation include chaos, global bifurcation theory and the role of symmetry, localised solutions (both in spectral and physical space), coupled oscillators and synchronisation, ergodic theory and stochastic dynamics, and pattern formation in fluid mechanics and reaction-diffusion systems. Developments in the basic theory and techniques of Nonlinear Dynamics go hand-in-hand with investigations of particular applications, such as fluid dynamics experiments, dynamics on complex networks and mixing in microfluidics.

Keywords: applied mathematics, numerical analysis, ordinary differential equations, method of lines, complex systems, networks