Geometrization of metric boundary data for Einstein's equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein's equations.Comment: Article to appear in Gen. Rel. Grav. volume in memory of Juergen Ehler

The Spin-2 Equation on Minkowski Background

The linearised general conformal field equations in their first and second order form are used to study the behaviour of the spin-2 zero-rest-mass equation on Minkowski background in the vicinity of space-like infinity.Comment: Contribution to the Proceedings of the Spanish Relativity Meeting ERE 2012, 4 page

Cosmological post-Newtonian expansions to arbitrary order

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter \ep=v_T/c (0<\ep < \ep_0), where $c$ is the speed of light, and $v_T$ is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M\cong [0,T)\times \Tbb^3, and converge as \ep \searrow 0 to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter \ep to any specified order with expansion coefficients that satisfy \ep-independent (nonlocal) symmetric hyperbolic equations

Existence of families of spacetimes with a Newtonian limit

J\"urgen Ehlers developed \emph{frame theory} to better understand the relationship between general relativity and Newtonian gravity. Frame theory contains a parameter $\lambda$, which can be thought of as $1/c^2$, where $c$ is the speed of light. By construction, frame theory is equivalent to general relativity for $\lambda >0$, and reduces to Newtonian gravity for $\lambda =0$. Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to study the Newtonian limit \ep \searrow 0 (i.e. $c\to \infty$). A number of ideas relating to frame theory that were introduced by J\"urgen have subsequently found important applications to the rigorous study of both the Newtonian limit and post-Newtonian expansions. In this article, we review frame theory and discuss, in a non-technical fashion, some of the rigorous results on the Newtonian limit and post-Newtonian expansions that have followed from J\"urgen's work

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.Comment: 20 page

New, efficient, and accurate high order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions

We construct new, efficient, and accurate high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators' spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions.Comment: 16 pages, 9 figures. The files with the coefficients for the derivative and dissipation operators can be accessed by downloading the source code for the document. The files are located in the "coeffs" subdirector

Stability Analysis of Galerkin/Runge-Kutta Navier-Stokes Discretisations on Unstructured Grids

This paper presents a timestep stability analysis for a class of discretisations applied to the linearised form of the Navier-Stokes equations on a 3D domain with periodic boundary conditions. Using a suitable definition of the `perturbation energy' it is shown that the energy is monotonically decreasing for both the original p.d.e. and the semi-discrete system of o.d.e.'s arising from a Galerkin discretisation on a tetrahedral grid. Using recent theoretical results concerning algebraic and generalised stability, sufficient stability limits are obtained for both global and local timesteps for fully discrete algorithms using Runge-Kutta time integration

Post-Newtonian expansions for perfect fluids

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations that have a first post-Newtonian expansion. The results here are based on the elliptic-hyperbolic formulation of the Einstein-Euler equations used in \cite{Oli06}, which contains a singular parameter \ep = v_T/c, where $v_T$ is a characteristic velocity associated with the fluid and $c$ is the speed of light. As in \cite{Oli06}, energy estimates on weighted Sobolev spaces are used to analyze the behavior of solutions to the Einstein-Euler equations in the limit \ep\searrow 0, and to demonstrate the validity of the first post-Newtonian expansion as an approximation

Interaction of laser radiation with the material during production powders and fibers

Воздействие лазерного излучения на твердое тело приводит к изменению температурного поля обрабатываемого вещества. Характер нагрева, определяющийся скоростями изменения температуры, температурных градиентов, оказывается различным в зависимости от свойств обрабатываемого материала и условий обработки. Основными физическими параметрами процесса лазерной обработки твердых тел являются удельная мощность поглощенного лазерного потока 104–109 Вт/см2 и время взаимодействия металла с лучом 10–5–10–8 с. При взаимодействии подобных импульсов излучения с поверхностью происходит мгновенное взрывоподобное плавление части материала и перевод окружающего поверхность вещества в плазменное состояние. Последующее расширение плазмы сопровождается возникновением ударной волны с пиковым давлением 1–10 ГПа, которая действует на материал, и имеет место диспергирование металла. Решена математическая задача нагрева и плавления цилиндрической пластины нормально падающим на ее поверхность световым потоком лазерного излучения, описываемая системой уравнений теплопроводности в трех сечениях нагреваемой пластины, которые характеризуются временным фактором воздействия лазерного излучения на вещество: 1) 0 ≤ t ≤ tm; 2) t > tm; 3) tm tm; 3) tm < t ≤ th (here tm, th is the time moment corresponding to the beginning of the formation of the liquid phase and the end of the melting of the plate, respectively). The calculated dependences of changes in the surface temperature of metal alloys X18N10T, X15N60 during the action of a laser radiation pulse with a duration of τ=5 ms are presented. The presence of a phase transition associated with metal melting (an inflection in the curves) leads to a temporary decrease in the rate of temperature growth. The distribution of temperature fields causes a significant heterogeneity in the distribution of temperature over the thickness of materials, which reaches 2000 °C or more depending on the thickness of the metal and the conditions of exposure. The temperature curves of the surface heating repeat the shape of the pulse, and the temperature of the rest of the metal has a nonlinear tendency to increase with the output to the asymptote. It is established that the process of explosive metal sputtering requires heating the volume of the material above the melting point at a thickness of 300–350 microns and an impact energy of 7–8 J. Reducing the level of energy impact to 5–6 J and increasing the thickness of the workpiece more than 500 microns does not provide the distribution of temperature fields required for the implementation of the spraying process

Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in √s = 7 TeV pp collisions with the ATLAS detector

A search for the direct production of charginos and neutralinos in final states with three electrons or muons and missing transverse momentum is presented. The analysis is based on 4.7 fb−1 of proton–proton collision data delivered by the Large Hadron Collider and recorded with the ATLAS detector. Observations are consistent with Standard Model expectations in three signal regions that are either depleted or enriched in Z-boson decays. Upper limits at 95% confidence level are set in R-parity conserving phenomenological minimal supersymmetric models and in simplified models, significantly extending previous results