21,059 research outputs found
Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space
This paper, Part I in a two-part series, presents (i) A simple and highly
efficient algorithm for evaluation of quasi-periodic Green functions, as well
as (ii) An associated boundary-integral equation method for the numerical
solution of problems of scattering of waves by doubly periodic arrays of
scatterers in three-dimensional space. Except for certain "Wood frequencies" at
which the quasi-periodic Green function ceases to exist, the proposed approach,
which is based on use of smooth windowing functions, gives rise to lattice sums
which converge superalgebraically fast--that is, faster than any power of the
number of terms used--in sharp contrast with the extremely slow convergence
exhibited by the corresponding sums in absence of smooth windowing. (The
Wood-frequency problem is treated in Part II.) A proof presented in this paper
establishes rigorously the superalgebraic convergence of the windowed lattice
sums. A variety of numerical results demonstrate the practical efficiency of
the proposed approach
Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data
In this paper we provide a reconstruction algorithm for piecewise-smooth
functions with a-priori known smoothness and number of discontinuities, from
their Fourier coefficients, posessing the maximal possible asymptotic rate of
convergence -- including the positions of the discontinuities and the pointwise
values of the function. This algorithm is a modification of our earlier method,
which is in turn based on the algebraic method of K.Eckhoff proposed in the
1990s. The key ingredient of the new algorithm is to use a different set of
Eckhoff's equations for reconstructing the location of each discontinuity.
Instead of consecutive Fourier samples, we propose to use a "decimated" set
which is evenly spread throughout the spectrum
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
Self-Consistent Cosmological Simulations of DGP Braneworld Gravity
We perform cosmological N-body simulations of the Dvali-Gabadadze-Porrati
braneworld model, by solving the full non-linear equations of motion for the
scalar degree of freedom in this model, the brane bending mode. While coupling
universally to matter, the brane-bending mode has self-interactions that become
important as soon as the density field becomes non-linear. These
self-interactions lead to a suppression of the field in high-density
environments, and restore gravity to General Relativity. The code uses a
multi-grid relaxation scheme to solve the non-linear field equation in the
quasi-static approximation. We perform simulations of a flat self-accelerating
DGP model without cosmological constant. The results of the DGP simulations are
compared with standard gravity simulations assuming the same expansion history,
and with DGP simulations using the linearized equation for the brane bending
mode. This allows us to isolate the effects of the non-linear self-couplings of
the field which are noticeable already on quasi-linear scales. We present
results on the matter power spectrum and the halo mass function, and discuss
the behavior of the brane bending mode within cosmological structure formation.
We find that, independently of CMB constraints, the self-accelerating DGP model
is strongly constrained by current weak lensing and cluster abundance
measurements.Comment: 21 pages; 10 figures. Revised version matching published versio
Critical Slowing-Down in Landau Gauge-Fixing Algorithms
We study the problem of critical slowing-down for gauge-fixing algorithms
(Landau gauge) in lattice gauge theory on a -dimensional lattice. We
consider five such algorithms, and lattice sizes ranging from to
(up to in the case of Fourier acceleration). We measure four
different observables and we find that for each given algorithm they all have
the same relaxation time within error bars. We obtain that: the so-called {\em
Los Alamos} method has dynamic critical exponent , the {\em
overrelaxation} method and the {\em stochastic overrelaxation} method have , the so-called {\em Cornell} method has slightly smaller than
and the {\em Fourier acceleration} method completely eliminates critical
slowing-down. A detailed discussion and analysis of the tuning of these
algorithms is also presented.Comment: 40 pages (including 10 figures). A few modifications, incorporating
referee's suggestions, without the length reduction required for publicatio
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