published_or_final_version / Mathematics / Master / Master of Philosophy

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A (3+1)-evolutionary method in the framework of Regge Calculus, known as "Parallelisable Implicit Evolutionary Scheme", is analysed and revised so that it accounts for causality. Furthermore, the ambiguities associated with the notion of time in this evolutionary scheme are addressed and a solution to resolving such ambiguities is presented. The revised algorithm is then numerically tested and shown to produce the desirable results and indeed to resolve a problem previously faced upon implementing this scheme. An important issue that has been overlooked in "Parallelisable Implicit Evolutionary Scheme" was the restrictions on the choice of edge lengths used to build the space-time lattice as it evolves in time. It is essential to know what inequalities must hold between the edges of a 4-dimensional simplex, used to construct a space-time, so that the geometry inside the simplex is Minkowskian. The only known inequality on the Minkowski plane is the "Reverse Triangle Inequality" which holds between the edges of a triangle constructed only from space-like edges. However, a triangle, on the Minkowski plane, can be built from a combination of time-like, space-like or null edges. Part of this thesis is concerned with deriving a number of inequalities that must hold between the edges of mixed triangles. Finally, the Raychaudhuri equation is considered from the point of view of Regge Calculus. The Raychaudhuri equation plays an important role in many areas of relativistic Physics and Astrophysics, most importantly in the proof of singularity theorems. An analogue to the Raychaudhuri equation in the framework of Regge Calculus is derived. Both (2+1)-dimensional and (3+1)-dimensional cases are considered and analogues for average expansion and shear scalar are found.

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Implicit large eddy simulation (ILES) has been shown, in the literature, to have some success for three-dimensional flow (e.g. see [Grinstein, F.F., Margolin, L.G. and Rider, W. Implicit Large Eddy Simulation. Cambridge, 2007]), but it has not previously been examined for two-dimensional flow. This thesis investigates whether ILES can be applied successfully to two-dimensional flow. Modified equation analysis is used to demonstrate the similarities between the truncation errors of certain numerical schemes and the subgrid terms of the barotropic vorticity equation (BVE). This presents a theoretical motivation for the numerical testing. Burgers equation is first used as a model problem to develop the ideas and methodology. Numerical schemes that are known to model Burgers equation well (shock capturing schemes) are shown to be implicitly capturing the subgrid terms of the one-dimensional inviscid Burgers equation through their truncation errors. Numerical tests are performed on three equation sets (BVE, Euler equations and the quasi-geostrophic potential vorticity equation) to assess the application of ILES to two-dimensional flow. The results for each of these equation sets show that the schemes considered for ILES are able to capture some of the subgrid terms through their truncation errors. In terms of accuracy, the ILES schemes are comparable (or outperform) schemes with simple explicit subgrid models when comparing vorticity solutions with a high resolution reference vorticity solution. The results suggest that conservation of vorticity is important to the successful application of ILES to two-dimensional flow, whereas conservation of momentum is not. The schemes considered for ILES are able to successfully model the downscale enstrophy transfer, but none of the schemes considered for ILES (or the schemes with simple subgrid models) can model the correct upscale energy transfer from the subgrid to the resolved scales. Energy backscatter models are considered and are used with the ILES schemes. It is shown that it is possible to create an energy conserving and enstrophy dissipating scheme, composed of an ILES scheme and a backscatter model, that improves the accuracy of the vorticity solution (when compared with the corresponding ILES scheme without backscatter).

Atmospheric mixing is a problem of exceptional importance and difficult to study. The anelastic approximation is the accepted fluid system governing the atmosphere over large vertical scales (about 8 km). The anelastic equations, unlike the Navier-Stokes equations, incorporate a nontrivial spatial divergence constraint on the velocity field. This yields a weakly compressible fluid flow. The basis of this study is to use numerical analysis to explore the effects of weak compressibility in the evolution of fluid governed by the anelastic equations, and the effects of incompressibility governed by the Navier-Stokes equation. The analysis then goes on to investigate the difference between three different initial conditions. Within each initial condition different density profiles are observed while varying parameters are investigated. Numerical results show that comparisons of incompressible Navier-Stokes equations to the anelastic fluid flow equations do not produce similar results. The weakly compressible flow creates a mixing barrier, stopping vertical fluid exchange. The perturbed middle region / US Navy (USN) author

Stellar feedback processes have been suggested as a mechanism for explaining various properties of galaxies, especially dwarf galaxies, which have weaker potentials and thus lower escape velocities for galactic winds. In this thesis, I present work done during my DPhil to better understand these processes. I begin by discussing the techniques used to simulate galaxies as collections of astrophysical fluids in a cosmolog- ical context, and present some methods for interpreting the results of such simulations. I then present two projects aimed at furthering our understanding of feedback in dwarf galaxies. The first project is the investigation of a suite of simulations of satellites of a Milky Way-class halo. We discuss the formation of high-redshift dwarf galaxies and the effect that supernova feedback and reionisation have on the gas content and star formation history of these objects. We find that neither process has a dramatic effect on the star formation rates in high redshift dwarf galaxies that have already begun forming stars prior to reionisation. We do find, however, that the population of satellites is dramatically altered by the presence of cooled gas in the host halo, which increases the tidal stripping of satellites that pass close to the host. The second project concerns detailed simulations of a 15 solar mass star throughout its evolution, studying photoionisation, wind and supernova feedback from this star in various environments. Preliminary results are given for these simulations, which are compared to the results of previous authors.

This thesis investigates the polynomial collocation method for the numerical solution of Cauchy type integral equations and the use of those equations and the related numerical techniques to solve two practical problem in Acoustics and Aerodynamics. Chapters I and II include the basic background material required for the development of the main body of the thesis. Chapter I discusses a number of practical problems which can be modelled as a singular integral equations. In Chapter II the theory of those equations is given in great detail. In Chapter III the polynomial collocation method for singular integral equations with constant coefficients is presented. A particular set of collocation points, namely the zeros of the first kind Chebyshev polynomials, is shown to give uniform convergence of the numerical approximation for the cases of the index K = 0. 1. The convergence rate for this method is also given. All these results were obtained under slightly stronger assumptions than the minimum required for the existence of an exact solution. Chapter IV contains a generalization of the results in Chapter III to the case of variable coefficients. In Chapter V an example of a practical problem which results in a singular integral equation and which is successfully solved by the collocation method is described in substantial detail. This problem consists of the interaction of a sound wave with an elastic plate freely suspended in a fluid. It can be modelled by a system of two coupled boundary value problems - the Helmholtz equation and the beam equation. The collocation method is then compared with asymptotic results and a quadrature method due to Miller. In Chapter VI an efficient numerical method is developed for solving problems with discontinuous right-hand sides. Numerical comparison with other methods and possible extensions are also discussed.