Computational Partial Differential Equations

A core activity in Applied Mathematics is the development of novel numerical and computational methods. Research in this area ranges from the theoretical analysis of numerical methods for surface representation or the solution of differential equations, to the development and practical application of novel algorithms in engineering, physics, biology and industrial design.

Boundary integral and spectral methods (Prof. Mark A. Kelmanson)

Free-surface viscous fluid mechanics problems arise naturally in many important areas of engineering an physical sciences. In recent decades, the boundary-integral method has proved to be a powerful tool in solving such problems. Prof. Kelmanson is interested in improving the accuracy of such methods by appealing to the techniques of spectral methods, particularly for problems in which rapid variations occur in the curvature of the free surface, so that surface-mesh dynamical adaption is also required. The ultimate aim is to develop general algorithms that are not restricted to any particular application.

Perturbation methods and algebraic manipulators (Prof. Mark A. Kelmanson)

A large number of realistically motivated problems born of science and engineering generally yield only to numerical methods. However, some yield to asymptotic analysis and perturbation methods, which, by their very construction, are highly complicated and cumbersome from an algebraic point of view. Many problems have already been conquered by automating this process using algebraic manipulators, particularly Reduce and Maple. Multi-purpose `exact' solvers for ODEs and PDEs are sought, particularly using both matched-asymptotic and multiple-scale expansions.

Inverse problems (Prof. Daniel Lesnic)

Whilst direct formulations consist of determining the effect of a given cause, in inverse formulations the situation is completely, or partially reversed. The interest is into the research of inverse problems for partial differential equations governing phenomena in fluid flow, elasticity, acoustics, heat transfer, mechanics of aerosols, etc. Typical practical applications relate to flows in porous media, heat conduction in materials, thermal barrier coatings, heat exchangers, corrosion, etc. The objectives are to investigate the existence, uniqueness and stability of the solution to the problem that mathematically models a physical phenomenon under investigation, and to develop new convergent, stable and robust algorithms for obtaining the desired solution. The analyses concern inverse boundary value problems, inverse initial value problems, parameter identification, inverse geometry and source determination problems.

Boundary element methods (Prof. Daniel Lesnic)

The Boundary Element Method (BEM) is attractive mainly due to the possibility of reducing the dimensionality of a boundary value problem described by linear partial differential equations. To be successful in the reduction of dimensionality it is needed to have the fundamental solution of the original partial differential equations available in an analytical or simple form. If this is not the case then of much interest is the development and computational implementation of recent BEMs such as the dual reciprocity, multiple reciprocity, analogous element, contour point, fundamental solutions and meshless methods for solving both direct and inverse problems of mathematical physics.

Spectral methods for computational fluid dynamics (Dr Evy Kersalé)

Advances in the understanding of the dynamics of fluids, e.g. in astrophysics and geophysics, rely strongly on the range of computational resources available. But even more importantly, the design of appropriate, fast, accurate and stable numerical schemes allows us to push fluids modelling to the limit of high performance computing, in terms of, e.g. physics, geometry and resolution.

Our prime interest lies in spectral methods, which have been applied with tremendous success to the numerical modelling of fluids, and particularly turbulent fluids, for they can achieve stunning accuracy in the solution of smooth problems. While their implementation is straightforward and their properties well understood in the case of simple models (e.g. periodic, smooth, Cartesian), a further analysis of spectral methods is still needed, e.g., for problems involving complex geometries or when their solutions can develop singularities.

We have more recently grown interest in spectral element methods (i.e. piecewise spectral methods) which offer a very promising approach to modelling fluids with non trivial geometries while ensuring exponential convergence. Such methods might also lead to a very efficient implementation of numerical codes based on spectral decomposition, global in essence, on distributed memory machines.

Ordinary differential equations (Dr Jitse Niesen)

While the group is focused on partial differential equations, part of our research concerns methods for solving ordinary differential equations. We study a wide variety of methods, not only classical ones such as Runge–Kutta methods but also so-called Geometric Integrators which respect the underlying geometric structure of the differential equations (e.g., symmetries or Hamiltonian structure). These methods can be used in their own right, but also to resolve the temporal derivatives in partial differential equations. For instance, exponential integrators seem to be a good candidate for time-stepping certain kinds of semi-linear partial differential equations.

Computational methods and design for industry (Prof. Michael J. Wilson)

The research of the design group is concerned with the mathematical description of the shape of physical objects for the purposes of design, analysis, and manufacture. The work is based upon an innovative approach to shape design for function, invented in Applied Mathematics at Leeds, called the PDE Method. In mathematical terms, surface generation is viewed as a 'boundary-value problem' in which information used to specify the shape of a surface is defined along its edges. In particular a designer specifies the shape of the edges of each patch, and the PDE Method produces a surface shape as a smooth 'blend' or transition between the edges. The mathematical representation used to generate each surface patch is the solution of a partial differential equation. The method has been used in a wide variety of application areas, including the design of ship hulls, marine propellers, engine components, and aircraft. A recent development is the shape parameterisation of biological systems, with dynamic representations of the human heart and biological membranes. The work has generated a significant level of external support from a wide range of organisations including EPSRC, the European Union, and NASA.