Lattice Green's Functions for High Order Finite Difference Stencils

Lattice Green's Functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2D or 3D LGFs in linear time, avoiding the need for brute-force multi-dimensional quadrature of numerically unstable integrals. Here we focus on higher-order discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with high-order dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson's equation on fully or partially unbounded 3D domains

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by multiply connected domains, nonconvex obstacles, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the obstacle, reducing its computational and implementation complexity. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, which provides the key to simulations with multiply connected domains and outflow boundaries. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains

Planning the Ranch for Greater Profit: A Study of Physical and Economic Factors Affecting Organization and Management of Ranches in the Edwards Plateau Grazing Area.

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First narrow-band search for continuous gravitational waves from known pulsars in advanced detector data

Spinning neutron stars asymmetric with respect to their rotation axis are potential sources of continuous gravitational waves for ground-based interferometric detectors. In the case of known pulsars a fully coherent search, based on matched filtering, which uses the position and rotational parameters obtained from electromagnetic observations, can be carried out. Matched filtering maximizes the signalto- noise (SNR) ratio, but a large sensitivity loss is expected in case of even a very small mismatch between the assumed and the true signal parameters. For this reason, narrow-band analysis methods have been developed, allowing a fully coherent search for gravitational waves from known pulsars over a fraction of a hertz and several spin-down values. In this paper we describe a narrow-band search of 11 pulsars using data from Advanced LIGO’s first observing run. Although we have found several initial outliers, further studies show no significant evidence for the presence of a gravitational wave signal. Finally, we have placed upper limits on the signal strain amplitude lower than the spin-down limit for 5 of the 11 targets over the bands searched; in the case of J1813-1749 the spin-down limit has been beaten for the first time. For an additional 3 targets, the median upper limit across the search bands is below the spin-down limit. This is the most sensitive narrow-band search for continuous gravitational waves carried out so far

Sensitivity of the Advanced LIGO detectors at the beginning of gravitational wave astronomy

The Laser Interferometer Gravitational Wave Observatory (LIGO) consists of two widely separated 4 km laser interferometers designed to detect gravitational waves from distant astrophysical sources in the frequency range from 10 Hz to 10 kHz. The first observation run of the Advanced LIGO detectors started in September 2015 and ended in January 2016. A strain sensitivity of better than 10−23/Hz−−−√ was achieved around 100 Hz. Understanding both the fundamental and the technical noise sources was critical for increasing the astrophysical strain sensitivity. The average distance at which coalescing binary black hole systems with individual masses of 30  M⊙ could be detected above a signal-to-noise ratio (SNR) of 8 was 1.3 Gpc, and the range for binary neutron star inspirals was about 75 Mpc. With respect to the initial detectors, the observable volume of the Universe increased by a factor 69 and 43, respectively. These improvements helped Advanced LIGO to detect the gravitational wave signal from the binary black hole coalescence, known as GW150914

Search for post-merger gravitational waves from the remnant of the binary neutron star merger GW170817

In Advanced LIGO, detection and astrophysical source parameter estimation of the binary black hole merger GW150914 requires a calibrated estimate of the gravitational-wave strain sensed by the detectors. Producing an estimate from each detector's differential arm length control loop readout signals requires applying time domain filters, which are designed from a frequency domain model of the detector's gravitational-wave response. The gravitational-wave response model is determined by the detector's opto-mechanical response and the properties of its feedback control system. The measurements used to validate the model and characterize its uncertainty are derived primarily from a dedicated photon radiation pressure actuator, with cross-checks provided by optical and radio frequency references. We describe how the gravitational-wave readout signal is calibrated into equivalent gravitational-wave-induced strain and how the statistical uncertainties and systematic errors are assessed. Detector data collected over 38 calendar days, from September 12 to October 20, 2015, contain the event GW150914 and approximately 16 of coincident data used to estimate the event false alarm probability. The calibration uncertainty is less than 10% in magnitude and 10 degrees in phase across the relevant frequency band 20 Hz to 1 kHz

First measurement of the Hubble Constant from a Dark Standard Siren using the Dark Energy Survey Galaxies and the LIGO/Virgo Binary–Black-hole Merger GW170814

International audienceWe present a multi-messenger measurement of the Hubble constant H 0 using the binary–black-hole merger GW170814 as a standard siren, combined with a photometric redshift catalog from the Dark Energy Survey (DES). The luminosity distance is obtained from the gravitational wave signal detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO)/Virgo Collaboration (LVC) on 2017 August 14, and the redshift information is provided by the DES Year 3 data. Black hole mergers such as GW170814 are expected to lack bright electromagnetic emission to uniquely identify their host galaxies and build an object-by-object Hubble diagram. However, they are suitable for a statistical measurement, provided that a galaxy catalog of adequate depth and redshift completion is available. Here we present the first Hubble parameter measurement using a black hole merger. Our analysis results in , which is consistent with both SN Ia and cosmic microwave background measurements of the Hubble constant. The quoted 68% credible region comprises 60% of the uniform prior range [20, 140] km s−1 Mpc−1, and it depends on the assumed prior range. If we take a broader prior of [10, 220] km s−1 Mpc−1, we find (57% of the prior range). Although a weak constraint on the Hubble constant from a single event is expected using the dark siren method, a multifold increase in the LVC event rate is anticipated in the coming years and combinations of many sirens will lead to improved constraints on H 0