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An investigation into the current methods employed to conserve vorticity in numerical calculations is undertaken. Osher’s flux for the artificial compressibility equations is derived, implemented and validated in Cranfield University’s second order finite volume compressible flow solver MERLIN. Characteristic Decomposition is applied as a method of vorticity conservation in both the compressible and artificial compressibility MERLIN solvers. The performance of this method for vorticity conservation in both these solvers is assessed. Following a discussion of the issues associated with application of limiter functions on unstructured grids three modified versions of the method of Characteristic Decomposition are proposed and tested in both the compressible and incompressible solvers. It is concluded that the method of Characteristic Decomposition is an effective method for improving vorticity conservation and compares favourably in terms of increased computational cost to vorticity conservation through grid refinement.

For a fully adaptive hp finite element programme to be implemented it is necessary to implement n-irregular meshes efficiently. This requires a sufficiently flexible data structure to be implemented. Because such flexibility is required, the traditional array based approach cannot be used because of its limited applicability. In this thesis this traditional approach has been replaced by an object orientated design and implementation. This leads to an implementation that can be extended easily and safely to include other problems for which it was not originally designed. The problems with maintaining continuity on such a diverse variety of meshes and how continuity is maintained are discussed. Then the main structure of the mesh is described in the form of domain, subdomains and elements. These are used in conjunction with constraint mappings to give a conforming approximation even with the most irregular of meshes. There are several varieties of matrix generated by the method each with its own problems of storage. Sparse matrices, with perhaps more than 95% of zero entries, need to be used along side dense matrices. In this thesis an object oriented matrix library is implemented that enables this variety of matrices to be used. An hp finite element algorithm is then implemented using the data structures, and is tested on a range of test problems. The method is shown to be effective on these problems.

This thesis is mainly focused on the regularity problem for the three-dimensional Navier-Stokes equations. \\\\ The three-dimensional freely decaying Navier-Stokes and Burgers equations are compared via direct numerical simulations, starting from identical {\it incompressible} initial conditions, with the same kinematic viscosity. From previous work by Kiselev and Ladyzenskaya (1957), the Burgers equations are known to be globally regular thanks to a maximum principle. In this comparison, the Burgers equations are split via Helmholtz decomposition with consequence that the potential part dominates over the solenoidal part. The nonlocal term $-{\bm u}\cdot\nabla p$ invalidates the maximum principle in the Navier-Stokes equations. Its probability distribution function and joint probability distribution functions with both energy and enstrophy are essentially symmetric with random fluctuations, which are temporally correlated in all three cases. We then evaluate nonlinearity depletion quantitatively in the enstrophy growth bound via the exponent $\alpha$ in the power-law $\frac{dQ}{dt}+2\nu P\propto(Q aP b) {\alpha}$, where $Q$ is enstrophy, $P$ is palinstrophy and $a$ and $b$ are determined by calculus inequalities. \\\\ Caffarelli-Kohn-Nirenberg theory defines a local Reynolds number over parabolic cylinder $Q_r$ as $\delta(r)=1/(\nu r)\int_{Q_r}|\nabla {\bm u}| 2\,d{\bm x}\,dt$. From this we determine a cross-over scale $r_* \propto L\left( \frac{ \overline{\|\nabla \bm{u} \| 2_{L 2}} } {\| \nabla \bm{u} \| 2_{L \infty}} \right) {1/3}$, corresponding to the change in scaling behavior of $\delta(r)$. Following the assumption that $E(k)\propto k {-q}$ $(1<q<3)$, it is shown that $r_*\propto \nu a$ where $a=\frac{4}{3(3-q)}-1$. Direct numerical simulations of isotropic turbulence with $R_{\lambda}\approx 100$ and random initial data result in the scaling $\delta(r)\propto r 4$, which extends {\it throughout the inertial range}. This follows from the smallness of the intermittency parameter $a\approx 0.26$. From this value, the $\beta$-model predicts a dissipation correlation exponent $\mu=\frac{4a}{1+a}\approx 0.8$ which is much larger than the experimental observations of $0.2-0.4$. This suggests that the $\beta$-model is valid qualitatively but not quantitatively. The scale $r_*$ gives a practical method for estimating intermittency. \\\\ By studying the steadily propagating shock wave solutions of the one-dimensional Burgers equation with passive scalar, we determine a relationship between the dissipation rate $\epsilon_\theta$ of passive scalar and Prandtl number $P_r$ as $\epsilon_\theta\propto1/\sqrt{P_r}$ for large $P_r$. The profile of the passive scalar manifests as a sum of $\tanh {2n+1}x$ for suitably scaled $x$ when $\nu\to 0$, implying that we must distinguish between different orders of the Heaviside function $H$ and $H n$. If we do not account for this, we obtain the incorrect relationship $\epsilon_\theta\propto1/P_r$. The correct evaluation of this dissipation anomaly therefore requires Colombeau's theory for multiplication of distributions.

This thesis considers structure preserving matrix methods for computations on Bernstein polynomials whose coefficients are corrupted by noise. The ill-posed operations of greatest common divisor computations and polynomial division are considered, and it is shown that structure preserving matrix methods yield excellent results. With respect to greatest common divisor computations, the most difficult part is the computation of its degree, and several methods for its determination are presented. These are based on the Sylvester resultant matrix, and it is shown that a new form of the Sylvester resultant matrix in the modified Bernstein basis yields the best results. The B´ezout resultant matrix in the modified Bernstein basis is also considered, and it is shown that the results from it are inferior to those from the Sylvester resultant matrix in the modified Bernstein basis.

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This thesis is concerned with the issue of finding an accurate, efficient and robust numerical solution technique for solving mathematical models of reactive flow. Two main issues of concern when solving these problems are large computational costs and numerical instabilities and inaccuracies. Over the past decades, many numerical techniques have been suggested for the solution of reacting flow problems. The work in this thesis is part of the continuing trend to find schemes which can solve reacting flow problems efficiently and robustly.

This thesis is concerned with the improvement of numerical methods, specifically boundaryelement methods (BEMs), for solving Fredholm integral equations in both one- and two dimensions. The improvements are based on novel (computer-algebra-based) error analyses that yield explicit forms of correction terms for a priori incorporation into BEM methods employing piecewise-polynomial interpolation in the numerical approximation. The work is motivated by the aim of reducing errors of BEM methods for low-degree interpolating polynomials, without significantly increasing the computational cost associated with higher-degree interpolation. The present thesis develops, implements and assesses improved BEMs on two fronts. First, a modified Nystr¨om method is developed for the solution of one-dimensional Fredholm integral equations of the second kind (FIE2s). The method is based upon optimal approximation and inclusion of an explicit form of orthogonal-polynomial integration error, and it can be extended to systems of integral equations. It is validated, in both the single and system cases, on challenging FIE2s that contain a finite number of (integrable) singularities, or points of limited differentiability, within the integral kernels. Second, BEMs are developed for solving two-dimensional FIE2s in the widely applicable context of harmonic boundary value problems in which the boundary conditions may be either continuous or discontinuous. In the latter case, modifying the BEM to conquer the adverse effect (on convergence with decreasing mesh size) caused by boundary singularities requires considerable additional theory and implementation; the motivation for doing so is that such singularities arise naturally in the modelling of, e.g., stress fractures in solid mechanics and dielectrics in electrostatics. For both non-singular and singular BVPs, standard BEMs are improved herein by optimal approximation and inclusion of explicit forms of Lagrange-interpolation integration errors. The modified BEMs are validated against pseudo-analytic results obtained by a conformal transformation method, for which a novel implementation of the inverse transformation (needed to recover the physical solution) is included explicitly by use of an algebraic manipulator. Through a set of test problems with known (or otherwise computable) solutions, both the one- and two-dimensional modified methods, for both regular and singular BVPs, are demonstrated to show marked improvements in performance over their unmodified counterparts.

In this thesis, we introduce and assess a new adaptive method for solving non-linear parabolic partial differential equations with fixed or moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is based upon the given initial data. For the moving boundary cases, the mesh movement at the boundary is governed by a second monitor function. The method is applied with different monitor functions, to the semilinear heat equation in one space dimension, and the porous medium equation in one and two space dimensions. The effects of optimising initial data for chosen monitors will be considered - in these cases, maintaining the initial distribution amounts to equidistribution. A quantification of the effects of a mesh moving away from an equidistribution are considered here, also the effects of tangling, and then untangling a mesh and restarting.

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