Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

Convergence of numerical time-averaging and stationary measures via Poisson equations

Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

A regularized iterative algorithm for limited-angle inverse Radon transform

The tomography problem is investigated when the available projections are restricted to a limited angular domain. It is shown that a previous algorithm proposed for extrapolating the data to the missing cone in Fourier space is unstable in the presence of noise because of the ill-posedness of the problem. A regularized algorithm is proposed, which converges to stable solutions. The efficiency of both algorithms is tested by means of numerical simulations. © 1983 Taylor and Francis Group, LLC.

Number of degrees of freedom in inverse diffraction

The problem of inverse diffraction from plane to plane is considered in the case where a finite aperture exists in the boundary plane. Singular values and singular functions for the problem are introduced, and the number of degrees of freedom is defined in terms of the distribution of the singular values. Numerical computations are presented for the one-dimensional problem, and it is shown that the effect of evanescent waves disappears at a distance of approximately one wavelength from the boundary plane, even when the dimension of the slit is comparable with the wavelength of the diffracted field. © 1983 Taylor & Francis Group, LLC.

Inversion of light scattering data by singular system analysis

We consider the problem of inverting experimental data obtained in light scattering experiments described by linear theories. We discuss applications to particle sizing and we describe fast and easy-to-implement algorithms which permit the extraction, from noisy measurements, of reliable information about the particle size distribution. © 1987, SPIE.

Linear inverse problems with discrete data: II. Stability and regularization

For pt.I. see ibid. vol.1, p.301 (1985). In the first part of this work a general definition of an inverse problem with discrete data has been given and an analysis in terms of singular systems has been performed. The problem of the numerical stability of the solution, which in that paper was only briefly discussed, is the main topic of this second part. When the condition number of the problem is too large, a small error on the data can produce an extremely large error on the generalised solution, which therefore has no physical meaning. The authors review most of the methods which have been developed for overcoming this difficulty, including numerical filtering, Tikhonov regularisation, iterative methods, the Backus-Gilbert method and so on. Regularisation methods for the stable approximation of generalised solutions obtained through minimisation of suitable seminorms (C-generalised solutions), such as the method of Phillips (1962), are also considered.

An application of quasi-Newton methods for the numerical solution of interface problems

Quasi-Newton methods are applied to solve interface problems which arise from domain decomposition methods. These interface problems are usually sparse systems of linear or nonlinear equations. We are interested in applying these methods to systems of linear equations where we are not able or willing to calculate the Jacobian matrices as well as to systems of nonlinear equations resulting from nonlinear elliptic problems in the context of domain decomposition. Suitability for parallel implementation of these algorithms on coarse-grained parallel computers is discussed.

Numerical modelling and validation of Marangoni and surface tension phenomena using the finite volume method

Surface tension induced flow is implemented into a numerical modelling framework and validated for a number of test cases. Finite volume unstructured mesh techniques are used to discretize the mass, momentum and energy conservation equations in three dimensions. An explicit approach is used to include the effect of surface tension forces on the flow profile and final shape of a liquid domain. Validation of this approach is made against both analytical and experimental data. Finally, the method is used to model the wetting balance test for solder alloy material, where model predictions are used to gain a greater insight into this process. Copyright © 2000 John Wiley & Sons, Ltd.

An experimental and numerical CFD study of turbulence in a tundish container

This paper describes work performed at IRSID/USINOR in France and the University of Greenwich, UK, to investigate flow structures and turbulence in a water-model container, simulating aspects typical of metal tundish operation. Extensive mean and fluctuating velocity measurements were performed at IRSID using LDA to determine the flow field and these form the basis for a numerical model validation. This apparently simple problem poses several difficulties for the CFD modelling. The flow is driven by the strong impinging jet at the inlet. Accurate description of the jet is most important and requires a localized fine grid, but also a turbulence model that predicts the correct spreading rates of jet and impinging wall boundary layers. The velocities in the bulk of the tundish tend to be (indeed need to be) much smaller than those of the jet, leading to damping of turbulence, or even laminar flow. The authors have developed several low-Reynolds number (low-Re) k–var epsilon model variants to compute this flow and compare against measurements. Best agreement is obtained when turbulence damping is introduced to account not only for walls, but also for low-Re regions in the bulk – the k–var epsilon model otherwise allows turbulence to accumulate in the container due to the restricted outlet. Several damping functions are tested and the results reported here. The k–ω model, which is more suited to transitional flow, also seems to perform well in this problem.

A volume of fluid numerical model for wave impacts at coastal structures

The first stages in the development of a new design tool, to be used by coastal engineers to improve the efficiency, analysis, design, management and operation of a wide range of coastal and harbour structures, are described. The tool is based on a two-dimensional numerical model, NEWMOTICS-2D, using the volume of fluid (VOF) method, which permits the rapid calculation of wave hydrodynamics at impermeable natural and man-made structures. The critical hydrodynamic flow processes and forces are identified together with the equations that describe these key processes. The different possible numerical approaches for the solution of these equations, and the types of numerical models currently available, are examined and assessed. Preliminary tests of the model, using comparisons with results from a series of hydraulic model test cases, are described. The results of these tests demonstrate that the VOF approach is particularly appropriate for the simulation of the dynamics of waves at coastal structures because of its flexibility in representing the complex free surfaces encountered during wave impact and breaking. The further programme of work, required to develop the existing model into a tool for use in routine engineering design, is outlined.

On a flexible elimination algorithm with applications to panel element equations

A flexible elimination algorithm is presented and is applied to the solution of dense systems of linear equations. Properties of the algorithm are exploited in relation to panel element methods for potential flow and subsonic compressible flow. Further properties in relation to distributed computing are also discussed.

A domain decomposition algorithm for inverse welding problems

A mathematical model and a numerical scheme for the inverse determination of heat sources generated by means of a welding process is presented in this paper. The accuracy of the heat source retrieval is discussed.

Numerical simulation of vacuum dezincing of lead alloy

Removing zinc by distillation can leave the lead bullion virtually free of zinc and also produces pure zinc crystals. Batch distillation is considered in a hemispherical kettle with water-cooled lid, under high vacuum (50 Pa or less). Sufficient zinc concentration at the evaporating surface is achieved by means of a mechanical stirrer. The numerical model is based on the multiphysics simulation package PHYSICA. The fluid flow module of the code is used to simulate the action of the stirring impeller and to determine the temperature and concentration fields throughout the liquid volume including the evaporating surface. The rate of zinc evaporation and condensation is then modelled using Langmuir’s equations. Diffusion of the zinc vapour through the residual air in the vacuum gap is also taken into account. Computed results show that the mixing is sufficient and the rate-limiting step of the process is the surface evaporation driven by the difference of the equilibrium vapour pressure and the actual partial pressure of zinc vapour. However, at higher zinc concentrations, the heat transfer through the growing zinc crystal crust towards the cold steel lid may become the limiting factor because the crystallization front may reach the melting point. The computational model can be very useful in optimising the process within its safe limits.