Numerical Analysis and Computational Modelling
Numerical analysis research in Reading is primarily focused on the numerical solution of differential equations.
Many physical phenomena can be modelled by differential equations, but – apart from some very specific cases – it is generally not possible to write down the solution to these problems in closed form. In order to understand the behaviour of the solution, it is thus often necessary to construct an approximation via a computational algorithm.
Our research concerns the development and analysis of such algorithms for a range of problems, with the goal, in general, being to achieve provably high accuracy without incurring excessive computational cost. Our group is also closely linked to the Data Assimilation and Inverse Problems group.
Our research group is focused on the following areas:
We have a large programme within numerical analysis in data assimilation and inverse problems, so large that it has spawned a separate research group within the Department, a 30-strong interdisciplinary research centre (Data Assimilation Research Centre), of staff from Mathematics and Statistics, from Meteorology, and staff from a 20-strong group of Met Office scientists based on the campus. One of our professors in Mathematics and Statistics – Roland Potthast – also leads data assimilation research at the German Met Office.
Our work on inverse problems also includes a large strand on applications in inverse scattering – developing algorithms using measurements of scattered acoustic and electromagnetic waves to detect the position, location and type of objects.
Boundary integral equation methods involve reformulating a partial differential equation on a domain as an integral equation on the boundary of the domain. This reduces the dimension of the problem, and can turn problems on unbounded domains into problems on bounded domains.
This approach leads to non-local integral operators, and the efficient solution of the resulting integral equations is a difficult task resulting in many mathematical and numerical challenges. These range from accurate computation of system matrix entries and efficient solution of the system of linear equations to error control and convergence theory.
One current emphasis is on developing methods for numerical solution of problems of rough surface scattering, arising out of which we have been developing boundary integral equation method-based algorithms for free surface problems, and methods for computing the spectra of general bounded linear operators.
Hybrid numerical-asymptotic boundary integral methods for high frequency scattering (Simon Chandler-Wilde)
The computing time for standard numerical methods for wave scattering problems grows rapidly as the frequency of the wave increases (equivalently as the size of the wavelength, relative to the scattering obstacle, decreases). This renders the solution to many problems of practical interest impossible with current technology.
Our group is at the forefront of the development and analysis of boundary integral equation methods for scattering problems that have the property that the computational time does not grow significantly as the frequency of the incident wave increases. The key idea of our approach is to incorporate knowledge of the oscillatory behaviour of the solution on the boundary directly into our approximation space, and to do this requires new results on high frequency asymptotics.
Work at Reading in this area has been supported by EPSRC (grant EP/F067798/1), the Leverhulme Trust, the Royal Society, the European Union, Arup Acoustics, BAE Systems, the BBC, the Institute for Cancer Research, the Met Office and Schlumberger.
We are interested in the stability and convergence analysis of different discretisation schemes for partial differential equations and in the underlying approximation theory. High-order finite element methods, based on using high- or variable-order piecewise polynomial finite element approximation, are a particularly elegant tool for the efficient numerical solution of partial differential equations, allowing for an effectual numerical approximation achieving high accuracy whilst keeping the number of unknowns relatively small.
A particular application area of interest is the propagation and the interaction of acoustic, electromagnetic and elastic waves in the time-harmonic regime, for which Trefftz-discontinuous Galerkin methods, a family of finite element methods which conjugate the power and flexibility of discontinuous Galerkin schemes with enhanced accuracy properties due to the use of wave-based discrete spaces, are particularly appropriate.
Related work includes the development and analysis of some new sign-definite variational formulations of time-harmonic boundary value problems. Work at Reading in this area has been supported by NERC, the Swiss National Science Foundation and Schlumberger.
The main focus of this work is on the design of consistent (mimetic) schemes for phase field and other evolution problems. This is particularly important and challenging in some models, for example the Navier-Stokes-Korteweg system, due to the delicate balance between diffusion and dispersion. Upon formulating a 'naive' discretisation of the problem, one may introduce high order perturbations which can lead to spurious behaviour and should not be neglected.
In addition we are interested in aposteriori error analysis and adaptivity of finite element methods for time dependent problems and discrete variational calculus for generic minimisation problems to derive discrete counterparts to Noether's theorem.
We are also working on the design of Galerkin schemes for nonvariational elliptic problems, with the particular application of providing a scheme for the linearisation of a fully nonlinear elliptic equation (like the Monge-Ampere problem) or a generic quasilinear one (the infinity Laplacian for example).
Adaptive moving mesh methods for time-dependent nonlinear partial differential equations (Mike Baines)
The main application areas of this work are the many and various moving boundary problems and blow-up problems that occur in physics and biology. Nonlinearities in the problems trigger a wealth of interesting behaviour which has been studied in much detail by mathematicians in the past.
For problems in which boundaries need to be tracked or resolution requirements vary with time, moving meshes are a natural choice. However, there is no information in the equations themselves about the mesh movement strategy any more than any other numerical approximation. The choice of numerical method therefore comprises a discretisation of the whole problem including the mesh. The consequent flexibility can be used to advantage. It may be argued that any numerical method should try to reproduce as many key features of a problem as possible, including the invariants and similarity properties inherent in the problem: this tenet is one of the cornerstones of geometric integration. We are interested in mesh movement strategies which do this with a minimum of fuss.
Confidence in a numerical method requires evidence that goes beyond these necessary conditions. On a numerical level the concepts of accuracy, stability and convergence have to be secured. The aims of research into numerical methods in this area at Reading are therefore 'accuracy upfront' with invariants built in as far as possible.
Conservation laws arise when some (physical) quantity is conserved, for example mass or momentum. Thus they are frequently used to model physical processes which involve movement of some medium, for example air, water or even road traffic. A distinguishing feature of conservation laws is that they allow the formation of shocks, i.e. discontinuous solutions, such as sonic booms, bores in rivers or traffic jams.
Since, in general, conservation laws must be solved numerically, these discontinuous solutions present an additional challenge in numerical modelling as they can trigger instabilities in the more classical numerical schemes. This has generated the need for the design of adaptive (non-linear) schemes which avoid such failings, together with associated techniques for their implementation in complex situations.
We are working on the numerical modelling of problems arising in computational neurosciences as part of interdisciplinary work within the Centre for Integrative Neuroscience and Neurodynamics. We are interested in the modelling and the analysis of the spatiotemporal evolution of neural tissue activity, in particular in the direct and inverse problems arising in 'neural field theory'.
Several integro-differential models are available in this area and we address some of the open questions related to their validation, analysis and numerical treatment.
+44 (0) 118 378
|Professor Michael Baines||Emeritus Professor, Sessional Lecturer||8993||m.j.baines|
|Professor Simon Chandler-Wilde||Professor of Applied Mathematics||6017||s.n.chandler-wilde|
|Dr Sarah Dance||Associate Professor of Data Assimilation||6452||s.l.dance|
|Dr Amos Lawless||Lecturer||5018||a.s.lawless|
|Professor Nancy Nichols||Professor of Applied Mathematics||8988||n.k.nichols|
|Dr Roland Potthast||Professor of Applied Mathematics||7614||r.w.e.potthast|
|Dr Peter K Sweby||Associate Professor||8675||p.k.sweby|
Former group members:
Dr Stephen Langdon (Associate Professor)
Dr Tristan Pryer (Associate Professor)
Dr Andrea Moiola (Senior Research Fellow)
Dr Alexey Chernov (Associate Professor)
Professor Sebastian Reich
Publications of the members of the research group, available at Reading Publications Database Centaur, can by found by clicking the link below.