1,916 research outputs found
Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension
We present a fast direct solver for two dimensional scattering problems,
where an incident wave impinges on a penetrable medium with compact support. We
represent the scattered field using a volume potential whose kernel is the
outgoing Green's function for the exterior domain. Inserting this
representation into the governing partial differential equation, we obtain an
integral equation of the Lippmann-Schwinger type. The principal contribution
here is the development of an automatically adaptive, high-order accurate
discretization based on a quad tree data structure which provides rapid access
to arbitrary elements of the discretized system matrix. This permits the
straightforward application of state-of-the-art algorithms for constructing
compressed versions of the solution operator. These solvers typically require
work, where denotes the number of degrees of freedom. We
demonstrate the performance of the method for a variety of problems in both the
low and high frequency regimes.Comment: 18 page
GRMHD in axisymmetric dynamical spacetimes: the X-ECHO code
We present a new numerical code, X-ECHO, for general relativistic
magnetohydrodynamics (GRMHD) in dynamical spacetimes. This is aimed at studying
astrophysical situations where strong gravity and magnetic fields are both
supposed to play an important role, such as for the evolution of magnetized
neutron stars or for the gravitational collapse of the magnetized rotating
cores of massive stars, which is the astrophysical scenario believed to
eventually lead to (long) GRB events. The code is based on the extension of the
Eulerian conservative high-order (ECHO) scheme [Del Zanna et al., A&A 473, 11
(2007)] for GRMHD, here coupled to a novel solver for the Einstein equations in
the extended conformally flat condition (XCFC). We fully exploit the 3+1
Eulerian formalism, so that all the equations are written in terms of familiar
3D vectors and tensors alone, we adopt spherical coordinates for the conformal
background metric, and we consider axisymmetric spacetimes and fluid
configurations. The GRMHD conservation laws are solved by means of
shock-capturing methods within a finite-difference discretization, whereas, on
the same numerical grid, the Einstein elliptic equations are treated by
resorting to spherical harmonics decomposition and solved, for each harmonic,
by inverting band diagonal matrices. As a side product, we build and make
available to the community a code to produce GRMHD axisymmetric equilibria for
polytropic relativistic stars in the presence of differential rotation and a
purely toroidal magnetic field. This uses the same XCFC metric solver of the
main code and has been named XNS. Both XNS and the full X-ECHO codes are
validated through several tests of astrophysical interest.Comment: 18 pages, 9 figures, accepted for publication in A&
Frequency-domain algorithm for the Lorenz-gauge gravitational self-force
State-of-the-art computations of the gravitational self-force (GSF) on
massive particles in black hole spacetimes involve numerical evolution of the
metric perturbation equations in the time-domain, which is computationally very
costly. We present here a new strategy, based on a frequency-domain treatment
of the perturbation equations, which offers considerable computational saving.
The essential ingredients of our method are (i) a Fourier-harmonic
decomposition of the Lorenz-gauge metric perturbation equations and a numerical
solution of the resulting coupled set of ordinary equations with suitable
boundary conditions; (ii) a generalized version of the method of extended
homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)] used to circumvent
the Gibbs phenomenon that would otherwise hamper the convergence of the Fourier
mode-sum at the particle's location; and (iii) standard mode-sum
regularization, which finally yields the physical GSF as a sum over regularized
modal contributions. We present a working code that implements this strategy to
calculate the Lorenz-gauge GSF along eccentric geodesic orbits around a
Schwarzschild black hole. The code is far more efficient than existing
time-domain methods; the gain in computation speed (at a given precision) is
about an order of magnitude at an eccentricity of 0.2, and up to three orders
of magnitude for circular or nearly circular orbits. This increased efficiency
was crucial in enabling the recently reported calculation of the long-term
orbital evolution of an extreme mass ratio inspiral [Phys. Rev. D {\bf 85},
061501(R) (2012)]. Here we provide full technical details of our method to
complement the above report.Comment: 27 pages, 4 figure
A bibliography on parallel and vector numerical algorithms
This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also
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