Spectral Element Methods for Astrophysical and Geophysical Fluid Dynamics (Astrophysical and Geophysical Fluid Dynamics)

Spectral methods are global numerical methods used to solve ODEs and PDEs, whereby the solution is approximated using a linear combination of functions solutions of Sturm-Liouville problems (usually polynomials or trigonometric polynomials). Spectral methods, which can achieve stunning accuracy in the solution of smooth problems, have been applied with tremendous success to the numerical modelling of fluids, and particularly turbulent fluids. The recent development of spectral element methods, which are piecewise spectral methods, offers a very promising approach to modelling fluids with non trivial geometries while still ensuring exponential convergence (in contrast with finite element methods). Such methods might also lead to very a efficient implementation of numerical codes based on spectral decomposition, global in essence, on distributed memory machines. Projects would involve a theoretical analysis of spectral element methods, some research on efficient implementations of linear solvers, and the actual application of the methods to problems motivated by astrophysical fluid dynamics.

keywords: applied mathematics, computational method, spectral element method, astrophysical fluid dynamics

Astrophysical and Geophysical Fluid Dynamics
The group in Leeds is one of the leading groups in the field of Astrophysical and Geophysical Fluid Dynamics, with international reputation in dynamo theory, astrophysical MHD and convection. The strength of the group is recognised by the award of several prizes and special fellowships. The group also holds one of the largest grants ever awarded to the University of Leeds. The nine permanent members of staff work with eighteen postdocs and postgraduate students.

The group is actively engaged in research in a wide-range of areas of astrophysical and geophysical fluid dynamics: from planetary dynamics (the geodynamo and planetary dynamos) through solar, stellar and galactic dynamics to highly compressible and relativistic dynamics on the largest scales. Magnetic fields are a strong theme, and the group is interested in how planets (like the Earth), stars (like the Sun), neutron stars, black holes and galaxies generate their magnetic fields through dynamo action. On the Sun, the well-known eleven-year sunspot cycle is a manifestation of the solar dynamo; indeed the solar magnetic field underlies all solar magnetic phenomena such as solar flares, coronal mass ejections and the solar wind. In the Earth, magnetic fields are generated by convection in the molten iron core, and it has recently become possible to solve the fundamental equations that govern the motion of fluids and the generation of magnetic fields, and successfully reproduce many of the observed features of the geomagnetic field. At the other end of the scale, magnetic fields are implicated in the formation of spectacular jets coming from neutron stars, black holes and galaxies. Without magnetic fields, the group has interests in waves and hydrodynamic instabilities in rotating stratified fluids, with applications to the Earth's atmosphere and ocean (and with application to other planets).