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This thesis is concerned with the numerical approximation of problems of fluid flow, in particular the stationary advection diffusion reaction equations and the time dependent, coupled equations of incompressible miscible displacement in a porous medium. We begin by introducing the continuous discontinuous Galerkin method for the singularly perturbed advection diffusion reaction problem. This is a method which coincides with the continuous Galerkin method away from internal and boundary layers and with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. We then turn our attention to the equations of incompressible miscible displacement for the concentration, pressure and velocity of one fluid in a porous medium being displaced by another. We show a reliable a posteriori error estimator for the time dependent, coupled equations in the case where the solution has sufficient regularity and the velocity is bounded. We remark that these conditions may not be attained in physically realistic geometries. We therefore present an abstract approach to the stationary problem of miscible displacement and investigate an a posteriori error estimator using weighted spaces that relies on lower regularity requirements for the true solution. We then return to the continuous discontinuous Galerkin method. We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We then show that by varying the penalization parameter on only a subset of the domain we reach the continuous discontinuous method in the limit. We present numerical experiments illustrating this approach both for equations of non-negative characteristic form (closely related to advection diffusion reaction equations) and to the problem of incompressible miscible displacement. We show that we may practically determine appropriate discontinuous and continuous regions, resulting in a significant reduction of the number of degrees of freedom required to approximate a solution, by using the properties of the discontinuous Galerkin approximation to the advection diffusion reaction equation. We finally present novel code for implementing the continuous discontinuous Galerkin method in C++.

We study the consequences of the applications of the 'Bootstrap’ hypothesis to the Unitary Symmetries. The groups SU(3), SU (6), U(6,6) and their applications to the strong Interactions of the Hadrons are discussed in the first Chapter. In the second Chapter, we discuss some of the methods that have been used in the past in dynamical (bootstrap) calculations. In the third Chapter, we consider the P-wave quark-pseudoscalar meson Octet scattering and investigate whether the existence of the three quarks, Q, which belong to the spinor representation of SU(3) and are supposed to have fractional charges, can be explained in a self-consistent scheme. The calculation shows that there exists a reciprocal bootstrap relationship between quarks, Q and some other particles, which have the baxyon number 1/3, spin 3/2 and belong to the 15-dimensional representation of SU(3), Using the determinantal method the self-consistent values we have obtained are: NA 2429Mev.,5251 Mev., 22 and 32, where MQ, MQ and g22 are respectively the masses and couplings of Q and Q*. In the fourth Chapter, we consider the meson-baryon scattering in the context of U(6,6) symmetry and study the mass-splittings of the , baryon Octet and Decouplet by N/D method. It is assumed that the SU(3) symmetry is approximately exact so that the masses of the baxyon Octet and Decouplet obtained by using the U(6,6) vertices in the calculation should correspond roughly to their respective 80(3) degenerate masses. Although the results are very much cut off-dependent, the calculations shows that by varying the cut off, S and U(6,6) coupling, g parameters It is possible to obtain the mass-spilttings in the right direction. Considering the very much involved nature of the calculations, one may conclude that the results agree reasonably well with the known experimental facts.

We describe the construction of the Veneziano model for strong interaction scattering amplitudes and its generalisation to multi-particle production processes, and examine some of its properties. In chapter one we introduce the ideas which form the input to the Veneziano model. In chapter two we study how they may all be simultaneously realised in a compact form and how we may extend the model to incorporate multiparticle production processes. We also indicate in this chapter how the model may be used in experimental applications, and what difficulties the simple model faces. In chapter three we examine how we may incorporate all the terms of the complete Veneziano expression into one term, provided certain trajectory constraints are satisfied. In chapter four we study an alternative five-point amplitude having a different structure to the conventional tree graph. In chapter five we extend this amplitude to the case with an arbitrary number of external lines.

This thesis is mainly concerned with conditional inference for contingency tables, where the MCMC method is used to take a sample of the conditional distribution. One of the most common models to be investigated in contingency tables is the independence model. Classic test statistics for testing the independence hypothesis, Pearson and likelihood chi-square statistics rely on large sample distributions. The large sample distribution does not provide a good approximation when the sample size is small. The Fisher exact test is an alternative method which enables us to compute the exact p-value for testing the independence hypothesis. For contingency tables of large dimension, the Fisher exact test is not practical as it requires counting all tables in the sample space. We will review some enumeration methods which do not require us to count all tables in the sample space. However, these methods would also fail to compute the exact p-value for contingency tables of large dimensions. \cite{DiacStur98} introduced a method based on the Grobner basis. It is quite complicated to compute the Grobner basis for contingency tables as it is different for each individual table, not only for different sizes of table. We also review the method introduced by \citet{AokiTake03} using the minimal Markov basis for some particular tables. \cite{BuneBesa00} provided an algorithm using the most fundamental move to make the irreducible Markov chain over the sample space, defining an extra space. The algorithm is only introduced for $2\times J \times K$ tables using the Rasch model. We introduce direct proof for irreducibility of the Markov chain achieved by the Bunea and Besag algorithm. This is then used to prove that \cite{BuneBesa00} approach can be applied for some tables of higher dimensions, such as $3\times 3\times K$ and $3\times 4 \times 4$. The efficiency of the Bunea and Besag approach is extensively investigated for many different settings such as for tables of low/moderate/large dimensions, tables with special zero pattern, etc. The efficiency of algorithms is measured based on the effective sample size of the MCMC sample. We use two different metrics to penalise the effective sample size: running time of the algorithm and total number of bits used. These measures are also used to compute the efficiency of an adjustment of the Bunea and Besag algorithm which show that it outperforms the the original algorithm for some settings.

Many theoretical treatments of quantum-mechanical scattering processes require the numerical solution of a set of second order ordinary differential equations of special form (with first derivative absent). The methods used to solve such sets of equations are generally based on step-by-step methods for solving a single second order differential equation over a fixed mesh. For example Chandra (1973) has published a computer program which uses de Vogelaere's method to solve the differential equations arising in a close-coupling formulation of quantum mechanical scattering problems. Chandra's program makes no attempt to monitor the local truncation error and leaves the choice of steplength strategy entirely to the user. Our aim is to improve on existing implementations of de Vogelaere's method for a single second order equation by incorporating a method of truncation error estimation and an automatic mesh-selection facility. Estimates of the truncation error in de Vogelaere's method are established together with an upper bound for the local truncation error; the interval of absolute stability is found to be [-2,0] and it is shown that the global truncation error is of order h(^4) where h is the steplength. In addition the characteristics of a method due to Raptis and Allison are investigated. A numerical comparison of computer programs which incorporate the methods of de Vogelaere, Numerov, Raptis and Allison and Adams-Bashforth Adams-Moulton, with an automatic error control is performed to determine which program gives the most reliable and efficient solution of the single channel radial Schrödinger equation. A modification of Chandra's program is provided which performs the numerical integration of a set of coupled second order homogeneous differential equations using de Vogelaere's method with an automatic error control.

Mathematical analysis has been undertaken for the vorticity formulation of the two dimensional Navier–Stokes equation on the β-plane with periodic boundary conditions. This equation describes the flow of fluid near the equator of the Earth. The long time behaviour of the solution of this equation is investigated and we show that, given a sufficiently regular forcing, the solution of the equation is nearly zonal. We use this result to show that, for sufficiently large β, the global attractor of this system reduces to a point. Another result can be obtained if we assume that the forcing is time-independent and sufficiently smooth. If the forcing lies in some Gevrey space, the slow manifold of the Navier–Stokes equation on the β-plane can be approximated with O(εn/2) accuracy for arbitrary n = 0, 1, · · · , as well as with exponential accuracy.

This thesis studies several constructions to produce aperiodic tilings with particular properties. The first chapter of this thesis gives a constructive method, exchanging edge to edge matching rules for a small atlas of permitted patches, that can decrease the number of prototiles needed to tile a space. We present a single prototile that can only tile R3 aperiodically, and a pair of square prototiles that can only tile R2 aperiodically. The thesis then details a construction that superimposes two unit square tilings to create new aperiodic tilings. We show with this method that tiling spaces can be constructed with any desired number of local isomorphism classes, up to (and including) an infinite value. Hyperbolic variants are also detailed. The final chapters of the thesis apply the concept of Toeplitz arrays to this construction, allowing it to be iterated. This gives a general method to produce new aperiodic tilings, from a set of unit square tilings. Infinite iterations of the construction are then studied. We show that infinite superimpositions of periodic tilings are describable as substitution tilings, and also that most Robinson tilings can be constructed by infinite superimpositions of given periodic tilings. Possible applications of the thesis are then briefly considered.

This thesis considers the development and implementation of a moving mesh strategy which is suitable for the numerical solution of partial differential equations (PDEs) that arise in problems relevant to meteorology. We concentrate primarliy on developing the Parabolic Monge-Ampère (PMA) moving mesh method. This is an r-adaptive method which is based on ideas from optimal transportation combined with the equidistribution principle applied to a (time varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method is straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension. Furthermore it produces meshes of provable regularity and smoothness. The mesh equation is augmented with suitable Neumann or periodic boundary conditions, with adaptivity along the boundaries handled automatically. Examples are presented of periodic and non-periodic meshes generated for a prescribed monitor function. The PMA mesh equation is then successfully coupled to a number of convection dominated PDEs in 1D and 2D and the relative merits of solving the resultant systems, using a simultaneous or an alternate solution procedure, are explored. The main test problem considered is the two-dimensional Eady problem, a meteorological problem which models the development of cyclones at mid-latitudes. Numerical solutions obtained on an adaptive grid using PMA are presented. The results show improved resolution of the front when compared to uniform grid solutions with an equivalent number of mesh points and computed with the same time step. A pressure-correction method is implemented on a semi-staggered adaptive grid that also conserves important physical properties of the solution. All numerical solutions presented involve discretising the underlying equations in the computational domain, which is fixed and uniform, using a finite difference scheme. An alternating strategy is implemented whereby the Eady equations are integrated first and then the mesh is updated. A conservative interpolation scheme is used to interpolate the updated solution from the old grid onto the new grid.

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