No description available.

No description available.

No description available.

Recently a dynamical systems approach to the process of transition to turbulence has arisen in which so called exact coherent structures play a key role. Such structures are equilibrium states which are exact solutions of the Navier-Stokes equations. Given the nature of the governing equations such states can be difficult to find due to the size of the state space and nonlinear effects. Techniques such as homotopy and artificial forcing have been used to overcome these issues. However, here we will focus on finding such states at large Reynolds numbers where an asymptotically large parameter allows the identification of the leading order effects, from which the structure of the states can be described. The states described in this thesis will be split into two general types. The first are of vortex-Tollmien-Schlichting-wave type and these states arise via the viscous instability mechanism. The flows studied here (plane Poiseuille flow and the asymptotic suction boundary layer) both exhibit a linear instability and hence we can find nonlinear travelling wave equilibrium states which bifurcate directly from the basic state. Importantly the class of states studied here have a large spanwise wavelength. We will find and investigate these states, finding that they appear to terminate in a singularity caused by the lack of spanwise diffusion at leading order. The second type are characterised by the presence of structures both in the freestream and close to the wall, and were first found by Deguchi & Hall (J. Fluid Mech, vol. 752, 2014, pp. 602-625). We will consider two variations of these states. The first is the effect of wall curvature which leads to the imperfect bifurcation of Görtler vortices from the basic state. The second is the spontaneous generation of near wall structures when freestream structures are assumed to appear impulsively at some downstream location.

This thesis investigates both theoretically and numerically the stability of travelling wave solutions using Fredholm determinants, on the real line. We identify a class of travelling wave problems for which the corresponding integral operators are of trace class. Based on the geometrical interpretation of the Evans function, we give an alternative proof connecting it to (modified) Fredholm determinants. We then extend that connection to the case of front waves by constructing an appropriate integral operator. In the context of numerical evaluation of Fredholm determinants, we prove the uniform convergence associated with the modified/regularised Fredholm determinants which generalises Bornemann's result on this topic. Unlike in Bornemann's result, we do not assume continuity but only integrability with respect to the second argument of the kernel functions. In support to our theory, we present some numerical results. We show how to compute higher order determinants numerically, in particular for integral operators belonging to classes I3 and I4 of the Schatten-von Neumann set. Finally, we numerically compute Fredholm determinants for some travelling wave problems e.g. the 'good' Boussinesq equation and the fth-order KdV equation.

We start this thesis with a numerical study of the convergence of the exponential time differencing (ETD) schemes and the semi-implicit Euler method for the Allen-Cahn equation and a reaction-convection-diffusion equation and also compare the accuracy and efficiency of these methods. Next, we solve the nonlinear convection-diffusion (green roof) model numerically using the ETD method and central difference approximation. This numerical solution is investigated for three different initial values for the saturation. Finally, we study travelling wave solutions and self-similar solutions for the green roof, in particular, for the two limiting cases of being close to a saturated region and a dry region. Travelling waves, in the form of fronts, are found for most realistic limiting values of saturation; travelling waves are also investigated for some limiting versions of the model. Self-similar solutions, valid for high or for low saturations, are additionally investigated.

The increasing complexity of modern hardware requires sophisticated programming techniques for programs to run efficiently. At the same time, increased power of modern hardware enables more advanced analyses to be included in compilers. This thesis focuses on one particular optimisation technique that improves utilisation of vector units. The foundation of this technique is the ability to chose memory mappings for data structures of a given program. Usually programming languages use a fixed layout for logical data structures in physical memory. Such a static mapping often has a negative effect on usability of vector units. In this thesis we consider a compiler for a programming language that allows every data structure in a program to have its own data layout. We make sure that data layouts across the program are sound, and most importantly we solve a problem of automatic data layout reconstruction. To consistently do this, we formulate this as a type inference problem, where type encodes a data layout for a given structure as well as implied program transformations. We prove that type-implied transformations preserve semantics of the original programs and we demonstrate significant performance improvements when targeting SIMD-capable architectures.

No description available.

In this work we have developed, implemented and tested a new multilevel hybrid algorithm for the adaptive finite element solution of a general class of variational problems. Our multilevel hybrid algorithm is a combination of node movement, edge swapping and local h-refinement. The adaptive strategy used in our hybrid algorithm is based upon the construction of a hierarchy of locally optimal meshes starting with a coarse grid for which the location and the connectivity of the nodes is optimised. The grid is then locally refined and the new mesh is optimised in the same manner. Our hybrid algorithm does not need any global solution of the problem, it uses only local information to update the nodal solution values by solving the local variational problems on a relatively small domain with only few unknowns. The node movement strategy is based upon knowledge of a steepest descent direction for each node found by a gradient calculation. A derivation of the gradient of stored energy with respect to the position of nodes is provided. A strategy for the movement of interior as well as boundary nodes is then given. Edge/face swapping in two and three space dimensions is explained and algorithms for node movement and edge swapping are given. Detailed descriptions of the possible local refinement strategies in two and three space dimensions are provided. Possible variants of our hybrid algorithm are considered and aspects of our hybrid algorithm regarding the quality of the meshes achieved and the computational work undertaken are discussed with some preliminary results. We have applied our hybrid algorithm on a number of test problems: considering linear, nonlinear and system of equations in two and three space dimensions. A detailed comparison of the results produced by our hybrid algorithm with other adaptive approaches has been made for all of our test problems. Results presented indicate that our hybrid algorithm can produce better meshes, in both two and three space dimensions, than is possible by more conventional adaptive strategies.

Inverse problems have become more and more important in various fields of science and technology, and have certainly been one of the fastest growing areas in applied mathematics over the last three decades. However, as inverse problems typically lead to mathematical models which are ill-posed, their solutions are unstable under data perturbations and classical numerical techniques fail to provide accurate and stable solutions. The work in thesis focuses on inverse force problems for the wave equation which consists of determining an unknown space/time-dependent force function acting on a vibrating structure from Cauchy boundary, final time displacement or integral data. The novel contribution of this thesis involves the development of efficient numerical algorithms for these inverse but ill-posed problems. We have used the boundary element method (BEM) to discretise the wave equation with a constant wave speed, and the finite difference method (FDM) for non-constant wave speed and/or inhomogeneous wave propagating medium. Imposing the available boundary and additional conditions, upon discretisation the inverse and ill-posed problem is recast into one of solving an ill- conditioned system of equations. The accuracy and convergence of the numerical results are investigated for various test force functions. The stability of the numerical solutions is investigated by introducing random noise into the input data which yields unstable results if no regularisation is used. The Tikhonov regularization method is employed in order to reduce the influence of the measurement errors on the numerical results. The choice of the regularization parameter is based on trial and error or on the L-curve criterion. Iterative regularizing methods such as the Landweber and conjugate gradient methods are also employed in one chapter. The inverse numerical solutions are compared with their known analytical solutions, where available, and with the corresponding direct numerical solutions otherwise.