In this thesis we study the neutron transport (Boltzmann transport equation) which is used to model the movement of neutrons inside a nuclear reactor. More specifically we consider the mono-energetic, time independent neutron transport equation. The neutron transport equation has predominantly been solved numerically by employing low order discretisation methods, particularly in the case of the angular domain. We proceed by surveying the advantages and disadvantages of common numerical methods developed for the numerical solution of the neutron transport equation before explaining our choice of using a discontinuous Galerkin (DG) discretisation for both the spatial and angular domain. The bulk of the thesis describes an arbitrary order in both angle and space solver for the neutron transport equation. We discuss some implementation issues, including the use of an ordered solver to facilitate the solution of the linear systems resulting from the discretisation. The resulting solver is benchmarked using both source and critical eigenvalue computations. In the pseudo three--dimensional case we employ our solver for the computation of the critical eigenvalue for three industrial benchmark problems. We then employ the Dual Weighted Residual (DWR) approach to adaptivity to derive and implement error indicators for both two--dimensional and pseudo three--dimensional neutron transport source problems. Finally, we present some preliminary results on the use of a DWR indicator for the eigenvalue problem.

Diffusion processes provide a natural way of modelling a variety of physical and economic phenomena. It is often the case that one is unable to observe a diffusion process directly, and must instead rely on noisy observations that are discretely spaced in time. Given these discrete, noisy observations, one is faced with the task of inferring properties of the underlying diffusion process. For example, one might be interested in inferring the current state of the process given observations up to the present time (this is known as the filtering problem). Alternatively, one might wish to infer parameters governing the time evolution the diffusion process. In general, one cannot apply Bayes’ theorem directly, since the transition density of a general nonlinear diffusion is not computationally tractable. In this thesis, we investigate a novel method of simplifying the problem. The stochastic differential equation that describes the diffusion process is replaced with a simpler ordinary differential equation, which has a random driving noise that approximates Brownian motion. We show how one can exploit this approximation to improve on standard methods for inferring properties of nonlinear diffusion processes.

No description available.

There are two main features in this thesis. The first is a study of the role of "tracts" of finite determination on the growth of integral functions. Such a study is of importance, since most of the known results in this field are based on the extreme case where the tracts of finite determination reduce to single lines. The second is the construction of functions, meromorphic and integral, bounded in tracts of positive angular measure, where the tracts are not confined to radial sectors.

Rearrangements are two measurable real-valued functions that have equal measure of pre-images of upper level sets. In this thesis, I will investigate several matters and problems relating to rearrangements: the relationship between assumptions on the measure space and desirable properties of the set of rearrangements, and the validity of rearrangement inequalities; generalising the Mountain Pass Lemma over rearrangements; and applying topological degree theory to boundary value problems involving rearrangements. From suppositions on the measure space, such as the measure space having finite measure and no atoms, it can proved that the set of rearrangements is contractible and locally contractible. The Mountain Pass Lemma over rearrangements can be generalised, so instead of considering continuous paths from the closed unit interval to the set of rearrangements; it will consider the continuous functions from the closed unit disc into the set of rearrangements. Topological degree theory is used to associate admissible triples of functions, sets and points with integers. These methods will be applied to a boundary value problem involving rearrangements, where the domain is almost equal to the union of balls, which has been studied using variational methods, providing new multiplicity results. The minimum number of solutions to this boundary value problem is found to be related exponentially to the number of balls contained in the domain.

This thesis considers one-dimensional moving mesh (MM) methods coupled with semi-Lagrangian (SL) discretisations of partial differential equations (PDEs) for meteorological applications. We analyse a semi-Lagrangian numerical solution to the viscous Burgers’ equation when using linear interpolation. This gives expressions for the phase and shape errors of travelling wave solutions which decay slowly with increasing spatial and temporal resolution. These results are verified numerically and demonstrate qualitative agreement for high order interpolants. The semi-Lagrangian discretisation is coupled with a 1D moving mesh, resulting in a moving mesh semi-Lagrangian (MMSL) method. This is compared against two moving mesh Eulerian methods, a two-step remeshing approach, solved with the theta-method, and a coupled moving mesh PDE approach, which is solved using the MATLAB solver ODE45. At each time step of the SL method, the mesh is updated using a curvature based monitor function in order to reduce the interpolation error, and hence numerical viscosity. This MMSL method exhibits good stability properties, and captures the shape and speed of the travelling wave well. A meteorologically based 1D vertical column model is described with its SL solution procedure. Some potential benefits of adaptivity are demonstrated, with static meshes adapted to initial conditions. A moisture species is introduced into the model, although the effects are limited.

The purpose of this work is to illuminate some of the non-smooth phenomena found in piecewise-smooth continuous and discrete dynamical systems, which do not occur in smooth systems. We will explain how such non-smooth phenomena arise in applications which experience impact, such as impact oscillators, and a type of rotating machine, called magnetic bearing systems. The study of their dynamics and sensitivity to parameter variation gives not just insights into the critical motion found in these applications, but also into the complexity and beauty in their own right. This work comprises two parts. The first part studies a general one-dimensional discontinuous power law map which can arise from impact oscillators with a repelling wall. Parameter variation and the influence of the exponent on the existence and stability of periodic orbits is presented. In the second part we analyse two coupled oscillators that model rotating machines colliding with a circular boundary under friction. The study of the dynamics of rigid bodies impacting with and without friction is approached in two ways. On the one hand existence and stability conditions for non-impacting and impacting invariant sets are derived using local and global methods. On the other hand the analysis of parameter variation reveals new non-smooth bifurcations. Extensive numerical studies confirm these results and reveal further phenomena not attainable otherwise.

We provide an L² theory for the local double Hilbert transform along an analytic surface (s, t ,φ(s, t )) in the Heisenberg group H¹, that is operator f ↦ Hφ f (x) := p.v.∫∣s∣,∣t∣≤1 f (x ∙ (s, t ,φ(s, t ))-¹) ds/s dt/t, where ∙ denotes the group operation in H1. This operator combines several features: it is amulti-parameter singular integral, its kernel is supported along a submanifold, and convolution is with respect to a homogeneous group structure. We reprove Hφ is always L²(H¹)→L²(H¹) bounded (a result first obtained in [Str12]) to illustrate the method and then refine it to characterize the largest class of polynomials P of degree less than d such that the operator HP is uniformly bounded when P ranges in the class. Finally, we provide examples of surfaces that can be treated by our method but not by the theory of [Str12].

This thesis is concerned with the study of the important effect of the gradient term in parabolic problems. More precisely, we study the global existence or nonexistence of solutions, and their asymptotic behaviour in finite or infinite time. Particularly when the power of the gradient term can increase to the power function of the solution. This thesis consists of five parts. (i) Steady-State Solutions, (ii) The Blow-up Behaviour of the Positive Solutions, (iii) Parabolic Liouville-Type Theorems and the Universal Estimates, (iv) The Global Existence of the Positive Solutions, (v) Viscous Hamilton-Jacobi Equations (VHJ). Under certain conditions on the exponents of both the function of the solution and the gradient term, the nonexistence of positive stationary solution of parabolic problems with gradient terms are proved in (i). In (ii), we extend some known blow-up results of parabolic problems with perturbation terms, which is not too strong, to problems with stronger perturbation terms. In (iii), the nonexistence of nonnegative, nontrivial bounded solutions for all negative and positive times on the whole space are showed for parabolic problems with a strong perturbation term. Moreover, we study the connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to parabolic problems with gradient terms. Namely, we use a general method for derivation of universal, pointwise a priori estimates of solutions from Liouville type theorems, which unifies the results of a priori bounds, decay estimates and initial and final blow up rates. Global existence and stability, and unbounded global solutions are shown in (iv) when the perturbation term is stronger. In (v) we show that the speed of divergence of gradient blow up (GBU) of solutions of Dirichlet problem for VHJ, especially the upper GBU rate estimate in n space dimensions is the same as in one space dimension.

This thesis discusses a two dimensional non-smooth dynamical system described by an autonomous ordinary differential equation. The right hand side of the differential equation is assumed to be discontinuous. We provide a local theory of existence, uniqueness and exponential asymptotic stability and state a formula for the basin of attraction. Our conditions are sufficient. Thetheory generalizes smooth dynamical systems theory by providing contraction conditions for two nearby trajectories at a jump. Such conditions have only previously been studied for a two dimensional nonautonomous differential equation. We provide an example of the theory developed in this thesis and show that we can determine stability of a periodic orbit without explicitly calculating it. This is the main advantage of our theory. Our conditions require to define a metric. This however, can turn out to be a difficult task, and at present, we do not have a method for finding such a metric systematically. The final part of this thesis considers an application of a nonsmooth dynamical system to biomechanics. We model an elderly person stepping over an obstacle. Our model assumes stiff legs, and suggests a gait strategy to overcome an obstacle. This work is in collaboration with Professor Wagner's research group at Institute for Sport Science at the University of Mϋnster. However, we only present work developed independently in this thesis.