Extreme Mass Ratio Inspirals: LISA's unique probe of black hole gravity

In this review article I attempt to summarise past and present-ongoing-work on the problem of the inspiral of a small body in the gravitational field of a much more massive Kerr black hole. Such extreme mass ratio systems, expected to occur in galactic nuclei, will constitute prime sources of gravitational radiation for the future LISA gravitational radiation detector. The article's main goal is to provide a survey of basic celestial mechanics in Kerr spacetime and calculations of gravitational waveforms and backreaction on the small body's orbital motion, based on the traditional `flux-balance' method and the Teukolsky black hole perturbation formalism.Comment: Invited review article, 45 pages, 23 figure

Gravitational waves in dynamical spacetimes with matter content in the Fully Constrained Formulation

The Fully Constrained Formulation (FCF) of General Relativity is a novel framework introduced as an alternative to the hyperbolic formulations traditionally used in numerical relativity. The FCF equations form a hybrid elliptic-hyperbolic system of equations including explicitly the constraints. We present an implicit-explicit numerical algorithm to solve the hyperbolic part, whereas the elliptic sector shares the form and properties with the well known Conformally Flat Condition (CFC) approximation. We show the stability andconvergence properties of the numerical scheme with numerical simulations of vacuum solutions. We have performed the first numerical evolutions of the coupled system of hydrodynamics and Einstein equations within FCF. As a proof of principle of the viability of the formalism, we present 2D axisymmetric simulations of an oscillating neutron star. In order to simplify the analysis we have neglected the back-reaction of the gravitational waves into the dynamics, which is small (<2 %) for the system considered in this work. We use spherical coordinates grids which are well adapted for simulations of stars and allow for extended grids that marginally reach the wave zone. We have extracted the gravitational wave signature and compared to the Newtonian quadrupole and hexadecapole formulae. Both extraction methods show agreement within the numerical errors and the approximations used (~30 %).Comment: 17 pages, 9 figures, 2 tables, accepted for publication in PR

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

Mathematical modelling of fixed bed reactors

Consideration is given to the solution of the highly exothermic fixed bed catalytic reactor problem taking into account heat and mass transfer resistances inside the catalyst pellets and across the external fluid film as well as radial temperature and oonoentration gradients in the fluid phase. Comparison of the model with the simpler quasi homogeneous repreaenation is made. In the region where the quasi homogeneous case predicts temperature "run-away", the added refinements assume some importance. Very significant; differences in behaviour are predicted. Indeed no temperature "run-away" is apparent. Inolucling simply a film mass and heat transfer resistance is no guarantee that temperature "run-away" will not be predicted. In fact, it is the particle diffusive resistance whioh is the main factor limiting the temperature effects. Since the region of temperature "run-away" is often in the practical range it is essential to use a more detailed model for design such as the one described here, especially if optimal operating conditions are being sought. Even on large digital computers, the computation time is excessively long if the sets of differential equations are solved simultaneously. By examining the intrapartiole equations in detail for a practical range of physical properties and operating conditions, it is shown that they may be reduced, to a lumped parameter form. While still retaining the characteristics of the general problem, the lumped parameter approximation can be solved in a substantially shorter time, thus taking its use in optimization and control studies feasible

Contract stresses in hip replacements.

SIGLEAvailable from British Library Document Supply Centre- DSC:D172337 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

VADER: A Flexible, Robust, Open-Source Code for Simulating Viscous Thin Accretion Disks

The evolution of thin axisymmetric viscous accretion disks is a classic problem in astrophysics. While models based on this simplified geometry provide only approximations to the true processes of instability-driven mass and angular momentum transport, their simplicity makes them invaluable tools for both semi-analytic modeling and simulations of long-term evolution where two- or three-dimensional calculations are too computationally costly. Despite the utility of these models, the only publicly-available frameworks for simulating them are rather specialized and non-general. Here we describe a highly flexible, general numerical method for simulating viscous thin disks with arbitrary rotation curves, viscosities, boundary conditions, grid spacings, equations of state, and rates of gain or loss of mass (e.g., through winds) and energy (e.g., through radiation). Our method is based on a conservative, finite-volume, second-order accurate discretization of the equations, which we solve using an unconditionally-stable implicit scheme. We implement Anderson acceleration to speed convergence of the scheme, and show that this leads to factor of $\sim 5$ speed gains over non-accelerated methods in realistic problems, though the amount of speedup is highly problem-dependent. We have implemented our method in the new code Viscous Accretion Disk Evolution Resource (VADER), which is freely available for download from https://bitbucket.org/krumholz/vader/ under the terms of the GNU General Public License.Comment: 58 pages, 13 figures, accepted to Astronomy & Computing; this version includes more discussion, but no other changes; code is available for download from https://bitbucket.org/krumholz/vader

Numerical methods for solving hyperbolic and parabolic partial differential equations

The main object of this thesis is a study of the numerical 'solution of hyperbolic and parabolic partial differential equations. The introductory chapter deals with a general description and classification of partial differential equations. Some useful mathematical preliminaries and properties of matrices are outlined. Chapters Two and Three are concerned with a general survey of current numerical methods to solve these equations. By employing finite differences, the differential system is replaced by a large matrix system. Important concepts such as convergence, consistency, stability and accuracy are discussed with some detail. The group explicit (GE) methods as developed by Evans and Abdullah on parabolic equations are now applied to first and second order (wave equation) hyperbolic equations in Chapter 4. By coupling existing difference equations to approximate the given hyperbolic equations, new GE schemes are introduced. Their accuracies and truncation errors are studied and their stabilities established. Chapter 5 deals with the application of the GE techniques on some commonly occurring examples possessing variable coefficients such as the parabolic diffusion equations with cylindrical and spherical symmetry. A complicated stability analysis is also carried out to verify the stability, consistency and convergence of the proposed scheme. In Chapter 6 a new iterative alternating group explicit (AGE) method with the fractional splitting strategy is proposed to solve various linear and non-linear hyperbolic and parabolic problems in one dimension. The AGE algorithm with its PR (Peaceman Rachford) and DR (Douglas Rachford) variants is implemented on tridiagonal systems of difference schemes and proved to be stable. Its rate of convergence is governed by the acceleration parameter and with an optimum choice of this parameter, it is found that the accuracy of this method, in general, is better if not comparable to that of the GE class of problems as well as other existing schemes. The work on the AGE algorithm is extended to parabolic problems of two and three space dimensions in Chapter 7. A number of examples are treated and the DR variant is used because of consideration of stability requirement. The thesis ends with a summary and recommendations for future work

Computing aspects of problems in non-linear prediction and filtering

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The fluid dynamics of pressure die casting processes

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The main text of the thesis consists of seven chapters. Following the literature survey, the work in Chapter 2 focuses on the study of the whole process of pressure die casting. The necessity of reclassification against the traditional 'three-stage' description is introduced in order to build a new basic tenet in constructing theoretical and practical investigations, which leads to the creation of fluid dynamic mathematical models of the process. On the basis of Chapter 2, Chapter 3 concentrates on dealing with the computer simulation of filling flow patterns occurring in the third stage that play the most important role in the process. The Simplified Marker And Cell (SMAC) method is used to obtain the computational results of the filling patterns of pressure die casting processes. On the basis of the computational analysis of typical examples, the viscosity, gravitational force and velocity effects on the overall filling pattern are examined, that lead to a validation of the important hypothesis that an ideal liquid can be used in numerical simulation of filling patterns and this enables one to achieve a more effective computer program for a complex cavity by quasi-3D or 3D models. Chapter 4 treats a specific problem of the residual flow that exists in the final (fifth) stage of the process. Mathematical models of residual flow are derived. Chapter 5 mainly consists of two parts. The first part deals with the application of similitude laws for simulating flows in pressure die casting processes. Detailed analyses and criteria on different relationships between model and prototype are given in order to correct previously offered formulae by Eckert (1989). The results of numerical simulation presented in Chapter 3 are also extended to validate the similitude criteria. The second part of Chapter 5 presents the use of a charge coupled device(CCD) for studying the diversity of fluid motion including the filling pattern, residual flow, thermals and air entrapment during cavity-fill and post cavity-fill within a one single shot cavity filling in water analogue experiments. A discussion, conclusions and suggestions for further study of the subject concerned are presented in Chapters 6 and Chapter 7