4,168 research outputs found
On the error propagation of semi-Lagrange and Fourier methods for advection problems
In this paper we study the error propagation of numerical schemes for the
advection equation in the case where high precision is desired. The numerical
methods considered are based on the fast Fourier transform, polynomial
interpolation (semi-Lagrangian methods using a Lagrange or spline
interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is
conservative and has to store more than a single value per cell).
We demonstrate, by carrying out numerical experiments, that the worst case
error estimates given in the literature provide a good explanation for the
error propagation of the interpolation-based semi-Lagrangian methods. For the
discontinuous Galerkin semi-Lagrangian method, however, we find that the
characteristic property of semi-Lagrangian error estimates (namely the fact
that the error increases proportionally to the number of time steps) is not
observed. We provide an explanation for this behavior and conduct numerical
simulations that corroborate the different qualitative features of the error in
the two respective types of semi-Lagrangian methods.
The method based on the fast Fourier transform is exact but, due to round-off
errors, susceptible to a linear increase of the error in the number of time
steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an
error growth that is proportional to the square root of the number of time
steps.
Finally, we show, for a simple model, that our conclusions hold true if the
advection solver is used as part of a splitting scheme.Comment: submitted to Computers & Mathematics with Application
Data analysis of gravitational-wave signals from spinning neutron stars. V. A narrow-band all-sky search
We present theory and algorithms to perform an all-sky coherent search for
periodic signals of gravitational waves in narrow-band data of a detector. Our
search is based on a statistic, commonly called the -statistic,
derived from the maximum-likelihood principle in Paper I of this series. We
briefly review the response of a ground-based detector to the
gravitational-wave signal from a rotating neuron star and the derivation of the
-statistic. We present several algorithms to calculate efficiently
this statistic. In particular our algorithms are such that one can take
advantage of the speed of fast Fourier transform (FFT) in calculation of the
-statistic. We construct a grid in the parameter space such that
the nodes of the grid coincide with the Fourier frequencies. We present
interpolation methods that approximately convert the two integrals in the
-statistic into Fourier transforms so that the FFT algorithm can
be applied in their evaluation. We have implemented our methods and algorithms
into computer codes and we present results of the Monte Carlo simulations
performed to test these codes.Comment: REVTeX, 20 pages, 8 figure
How to mesh up Ewald sums (I): A theoretical and numerical comparison of various particle mesh routines
Standard Ewald sums, which calculate e.g. the electrostatic energy or the
force in periodically closed systems of charged particles, can be efficiently
speeded up by the use of the Fast Fourier Transformation (FFT). In this article
we investigate three algorithms for the FFT-accelerated Ewald sum, which
attracted a widespread attention, namely, the so-called
particle-particle-particle-mesh (P3M), particle mesh Ewald (PME) and smooth PME
method. We present a unified view of the underlying techniques and the various
ingredients which comprise those routines. Additionally, we offer detailed
accuracy measurements, which shed some light on the influence of several tuning
parameters and also show that the existing methods -- although similar in
spirit -- exhibit remarkable differences in accuracy. We propose combinations
of the individual components, mostly relying on the P3M approach, which we
regard as most flexible.Comment: 18 pages, 8 figures included, revtex styl
Multiple Staggered Mesh Ewald: Boosting the Accuracy of the Smooth Particle Mesh Ewald Method
The smooth particle mesh Ewald (SPME) method is the standard method for
computing the electrostatic interactions in the molecular simulations. In this
work, the multiple staggered mesh Ewald (MSME) method is proposed to boost the
accuracy of the SPME method. Unlike the SPME that achieves higher accuracy by
refining the mesh, the MSME improves the accuracy by averaging the standard
SPME forces computed on, e.g. , staggered meshes. We prove, from theoretical
perspective, that the MSME is as accurate as the SPME, but uses times
less mesh points in a certain parameter range. In the complementary parameter
range, the MSME is as accurate as the SPME with twice of the interpolation
order. The theoretical conclusions are numerically validated both by a uniform
and uncorrelated charge system, and by a three-point-charge water system that
is widely used as solvent for the bio-macromolecules
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