15,762 research outputs found
Black hole evolution with the BSSN system by pseudo-spectral methods
We present a new pseudo-spectral code for the simulation of evolution systems
that are second order in space. We test this code by evolving a non-linear
scalar wave equation. These non-linear waves can be stably evolved using very
simple constant or radiative boundary conditions, which we show to be
well-posed in the scalar wave case. The main motivation for this work, however,
is to evolve black holes for the first time with the BSSN system by means of a
spectral method. We use our new code to simulate the evolution of a single
black hole using all applicable methods that are usually employed when the BSSN
system is used together with finite differencing methods. In particular, we use
black hole excision and test standard radiative and also constant outer
boundary conditions. Furthermore, we study different gauge choices such as
and constant densitized lapse. We find that these methods in principle
do work also with our spectral method. However, our simulations fail after
about due to unstable exponentially growing modes. The reason for this
failure may be that we evolve the black hole on a full grid without imposing
any symmetries. Such full grid instabilities have also been observed when
finite differencing methods are used to evolve excised black holes with the
BSSN system.Comment: 10 pages, 9 figure
Hyperboloidal slices for the wave equation of Kerr-Schild metrics and numerical applications
We present new results from two open source codes, using finite differencing
and pseudo-spectral methods for the wave equations in (3+1) dimensions. We use
a hyperboloidal transformation which allows direct access to null infinity and
simplifies the control over characteristic speeds on Kerr-Schild backgrounds.
We show that this method is ideal for attaching hyperboloidal slices or for
adapting the numerical resolution in certain spacetime regions. As an example
application, we study late-time Kerr tails of sub-dominant modes and obtain new
insight into the splitting of decay rates. The involved conformal wave equation
is freed of formally singular terms whose numerical evaluation might be
problematically close to future null infinity.Comment: 15 pages, 12 figure
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
We design an arbitrary high-order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
an equivalent flux differencing formulation of the split form of the equations.
We prove that this discretisation exactly preserves the local mass and
momentum. Furthermore, combined with a special numerical interface flux
function, the method exactly preserves the mathematical entropy, which is the
total energy for the shallow water equations. By adding a specific form of
interface dissipation to the baseline entropy conserving scheme we create a
provably entropy stable scheme. That is, the numerical scheme discretely
satisfies the second law of thermodynamics. Finally, with a particular
discretisation of the bathymetry source term we prove that the numerical
approximation is well-balanced. We provide numerical examples that verify the
theoretical findings and furthermore provide an application of the scheme for a
partial break of a curved dam test problem
Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing
We describe a modification of a fourth-order accurate ``moving puncture''
evolution code, where by replacing spatial fourth-order accurate differencing
operators in the bulk of the grid by a specific choice of sixth-order accurate
stencils we gain significant improvements in accuracy. We illustrate the
performance of the modified algorithm with an equal-mass simulation covering
nine orbits.Comment: 13 pages, 6 figure
Interpolation of nonstationary high frequency spatial-temporal temperature data
The Atmospheric Radiation Measurement program is a U.S. Department of Energy
project that collects meteorological observations at several locations around
the world in order to study how weather processes affect global climate change.
As one of its initiatives, it operates a set of fixed but irregularly-spaced
monitoring facilities in the Southern Great Plains region of the U.S. We
describe methods for interpolating temperature records from these fixed
facilities to locations at which no observations were made, which can be useful
when values are required on a spatial grid. We interpolate by conditionally
simulating from a fitted nonstationary Gaussian process model that accounts for
the time-varying statistical characteristics of the temperatures, as well as
the dependence on solar radiation. The model is fit by maximizing an
approximate likelihood, and the conditional simulations result in
well-calibrated confidence intervals for the predicted temperatures. We also
describe methods for handling spatial-temporal jumps in the data to interpolate
a slow-moving cold front.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS633 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Solving reaction-diffusion equations 10 times faster
The most popular numerical method for solving systems of reaction-diffusion equations continues to be a low order finite-difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high--order methods one can speed up such simulations for 2D and 3D problems by factors of 10--100. A short MATLAB code (2/3D) that can serve as a template is included.\ud
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This work was supported by the Engineering and Physical Sciences Research Council (UK) and by the MathWorks, Inc
PSpectRe: A Pseudo-Spectral Code for (P)reheating
PSpectRe is a C++ program that uses Fourier-space pseudo-spectral methods to
evolve interacting scalar fields in an expanding universe. PSpectRe is
optimized for the analysis of parametric resonance in the post-inflationary
universe, and provides an alternative to finite differencing codes, such as
Defrost and LatticeEasy. PSpectRe has both second- (Velocity-Verlet) and
fourth-order (Runge-Kutta) time integrators. Given the same number of spatial
points and/or momentum modes, PSpectRe is not significantly slower than finite
differencing codes, despite the need for multiple Fourier transforms at each
timestep, and exhibits excellent energy conservation. Further, by computing the
post-resonance equation of state, we show that in some circumstances PSpectRe
obtains reliable results while using substantially fewer points than a finite
differencing code. PSpectRe is designed to be easily extended to other problems
in early-universe cosmology, including the generation of gravitational waves
during phase transitions and pre-inflationary bubble collisions. Specific
applications of this code will be pursued in future work.Comment: 22 pages; source code for PSpectRe available:
http://easther.physics.yale.edu v2 Typos fixed, minor improvements to
wording; v3 updated as per referee comment
Pulsar timing analysis in the presence of correlated noise
Pulsar timing observations are usually analysed with least-square-fitting
procedures under the assumption that the timing residuals are uncorrelated
(statistically "white"). Pulsar observers are well aware that this assumption
often breaks down and causes severe errors in estimating the parameters of the
timing model and their uncertainties. Ad hoc methods for minimizing these
errors have been developed, but we show that they are far from optimal.
Compensation for temporal correlation can be done optimally if the covariance
matrix of the residuals is known using a linear transformation that whitens
both the residuals and the timing model. We adopt a transformation based on the
Cholesky decomposition of the covariance matrix, but the transformation is not
unique. We show how to estimate the covariance matrix with sufficient accuracy
to optimize the pulsar timing analysis. We also show how to apply this
procedure to estimate the spectrum of any time series with a steep red
power-law spectrum, including those with irregular sampling and variable error
bars, which are otherwise very difficult to analyse.Comment: Accepted by MNRA
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