

Project SummariesBelow are some of the projects currently under investigation in the School:
Modelling of Greenland and Antarctica Numerical Modelling of Greenland and AntarcticaDr. A. Payne, School of Geographical Sciences The aim is be able to predict the future evolution of the Greenland and Antarctic Ice Sheets. We use a variety of numerical (implicit finitedifference techniques) models to simulate the flow of these ice masses. They typically incorporate a prognostic equation ice thickness evolution (as a consequence of flow divergence and snow accumulation), the 3d advection/diffusion equation for temperature evolution, and a diagnostic equation for stress and velocity regimes. The models use the MPI protocol to run on parallel processors. CFD for Hypersonic PropulsionPeter Ess, Department of Aerospace Engineering. Blunt body generated combustion for supersonic (Mach 5) stoichiometrically premixed airhydrogen gas flow is studied numerically for different channel blockage ratios in order to find optimum conditions for combustion. Hence, the laminar NavierStokes equations with detailed chemical reaction schemes are used, yielding a very stiff system of equations. For this, an implicit multiblock finitevolume code is being developed and used to simulate such gas flows in steadystate or timeaccurate manner on single or multiprocessor computers. Transport and Stirring in Geophysical Flows: A Dynamical Systems ApproachDr. Ana Maria Mancho, Dr. D. Small, Prof. S. Wiggins, School of Mathematics, Department of Applied Mathematics Dynamical systems theory offers an ideal mathematical framework in which to study Lagrangian transport and mixing issues in geophysical flows. A code which simulates a 3 layer quasigeostrophic model of the ocean, is used to generate data where many ideas of transport are tested and developed. Different regimes are reached by changing the input parameters. These data are analyzed with several python codes which identificate stagnation points (SP) and calculate distinguished hyperbolic trajectories (DHT) near those points. The results are prepared to be visualized with TECPLOT. In order to calculate manifolds associated with the DHT we are analyzing effectiveness of different numerical interpolation schemes for tracking particle trajectories in non time periodic finite vector fields. Chaotic Wave ScatteringDr. Ishio Hiromu, School of Mathematics, Department of Applied Mathematics. I am solving the wave equation for a chaotic open cavity. Discretization of the equation reduces to a linear matrix equation which should be diagonalized to obtain its solution. I iterate the same calculation by changing an external parameter such as wave length etc. Therefore, parallel processing of the calculation is strongly desirable. Because of the complexity of the systems with chaotic geometries, highperformance computers are required to solve the wave equation of particles. We adopt plane wave expansion method, obtain large scale of matrices each element of which has integrals, and calculate the inverse of them. In most of the time, we iterate the same calculation, changing external parameters. Superconductivity in ZrZn_{2}Ben Powell, Department of Physics. ZrZn_{2} is a recently discovered ferromagnetic superconductor. The coexistence of ferromagnetism and superconductivity is a great surprise as normally superconductivity is destroyed by magnetic fields or magnetic impurities in the crystal. Early results show that for a particular choice of parameters the critical temperature of the superconductor rises as the exchange splitting of the ferromagnet increases. We will study the appropriate Bogoliubovde Gennes equations for this system. To do this involves diagonalising a matrix at a large number of points in reciprocal space and then integrating over all of the eigenstates independently at each of the points in reciprocal space. This is an intrinsically parallel task. The diagonalisation of the BdG matrices at each point in reciprocal space are independent of one another. Therefore the integration mesh can be subdivided into independent regions, each handled by a separate processor. Also each different temperature and magnetic field parameter set can be handled independently. Quantum Oscillations in the Superconducting State of MgB_{2}Dr. Gilles Santi, Department of Physics. The discovery of superconductivity in MgB_{2} with a T_{c} of 39 K has spawned a tremendous amount of experimental and theoretical work. From the various experiments, the emerging consensus seems to go towards an electronphonon driven, twogap superconductor. It is proposed to compute the damping of the quantum oscillations in the superconducting state using the semiclassical theory for superconductors and the result from abinitio electronic structure calculation. The pseudopotential plane wave electronic structure code is parallelized over the G vectors (the different plane waves). Ab Initio Molecular Dynamics of Hydrothermal SolutionsDr. D. Sherman, Earth Sciences. Hydrothermal and aqueous solutions play a fundamental role in geochemistry. The formation of ore deposits and the transport of metal pollutants in groundwater all depend on the coordination chemistry of metals in aqueous solutions. The determination of aqueous coordination chemistry, especially at high temperature, is difficult. Some of the most fundamental chemical systems in geology are still poorly understood. Ab initio molecular dynamical simulations using the Car Parinello algorithm and density functional theory allow us to probe the coordination chemistry of metals in aqueous solutions as a function of temperature for the first time. The CASTEP code for such calculations has been developed and parallelized by the UK Car Parinello consortium. We have used this code for work at the Manchester computing centre with promising results. For example, we have shown that the chemistry of copper is different from that predicted by extrapolated thermodynamic data. At the same time, our simulations are in perfect agreement with recent spectroscopic results. It is clear that atomistic calculations on ore forming solutions may greatly change our understanding of geochemical processes. Saturation of electrostatic instability in twospecies plasmaDr. R.R. Kerswell, School of Mathematics
A classical problem in plasma theory is how electrostatic instabilities develop
in the nonlinear regime. Traditional approaches treat the single species Vlasov
equation and predict a trapping scaling whereby the saturation amplitude scales
as gamma to the power 2 where gamma is the distance of a controlling parameter
from criticality. However, in fact, a plasma really consists of two species 
electrons and heavy ions. Analysis extended to include this second species
suggests that in this new situation nonlinearity becomes important when the
amplitude is O(gamma to the power 5 over 2). The project is fleshing out this
theory for the 2species Vlasov system and making quantitative comparisons with
numerical solutions of the full equations. These solutions possess fine structure
at particle resonances (critical layers) and hence are computationally expensive
to capture.
Parallel Solution of Sparse Linear Systems Defined Over GF(p)Dr. D Page, Department of Computer Science
The security of modern public key cryptography is usually based on the presumed
hardness of problems such as integer factorisation or the finding discrete
logarithms. As such, cryptosystems are generally parameterised so that it is
computationally infeasible for an attacker to solve said problems and hence
undo the security of the system. The Number Field Sieve (NFS) and Function
Field Sieve (FFS) are two methods that are able to attack the problems of
integer factorisation and finding discrete logarithms respectively, in a much
more efficient way than using a naive, brute force approach. These methods
operate in two phases. First, the sieving phase collects a large number of
relations that represent small pieces of information about the problem being
attacked. These relations are combined into a large linear system, whose
solution in the second, matrix phase, allows efficient calculation of answers
to the initial problem.
Electronic Structure of Weak Itinerant FerromagnetsDr. S. Dugdale, Department of Physics
Following the discovery of the coexistence of weak itinerant
ferromagnetism with superconductivity in UGe2, ZrZn2 and URhGe, there
has been a great deal of interest in understanding their electronic
structure. These materials are likely to have exotic pairing mechanisms,
with magnetic fluctutations being a likely candidate.
Particularly when combined with experimental investigations,
firstprinciples calculations of this electronic structure
can provide significant insight into this behaviour.
CFD for Aeroelastic SimulationDr. C.B. Allen, Department of Aerospace Engineering
Numerical simulation of aeroelastic response involves
timesynchronised simulation of unsteady fluid flows and
structural response, over long integration times, to
compute the timedependent response of flexible structures.
Furthermore, numerical dissipation present in all CFD codes
is grid density dependent and, hence, to capture the
essential physics of these flows, large meshes are required,
in the region of 0.55.0 million points for a fixedwing case.
Parallel processing is, therefore, essential for these flows,
both to reduce runtimes and distribute the required memory.
Vortex Interactions at HighReynolds Number With Meshless Methods.Dr. L.A. Barba, Department of Applied Mathematics
Computing the interaction of viscous vortices using traditional
CFD methods is severely hindered by numerical diffusion. In
general, the vortices diffuse too rapidly to properly capture the
details of their interaction with each other or with structures.
To deal with this problem, a fully meshless method has been
developed, which is characterized by nondiffusive truncation
errors. It is a new formulation of the vortex particle method,
using the core spreading scheme for viscous effects, and a
meshless spatial adaption technique based on radial basis function
(RBF) interpolation. Numerical experiments have demonstrated
increased accuracy in comparison with the standard approach of
remeshing with highorder kernels. The method has been implemented
in parallel using the
PETSc library.




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