984 research outputs found
A multidimensional hydrodynamic code for structure evolution in cosmology
A cosmological multidimensional hydrodynamic code is described and tested.
This code is based on modern high-resolution shock-capturing techniques. It can
make use of a linear or a parabolic cell reconstruction as well as an
approximate Riemann solver. The code has been specifically designed for
cosmological applications. Two tests including shocks have been considered: the
first one is a standard shock tube and the second test involves a spherically
symmetric shock. Various additional cosmological tests are also presented. In
this way, the performance of the code is proved. The usefulness of the code is
discussed; in particular, this powerful tool is expected to be useful in order
to study the evolution of the hot gas component located inside nonsymmetric
cosmological structures.Comment: 34 pages , LaTex with aasms4.sty, 7 postscript figures, figure 4
available by e-mail, tared , gziped and uuencoded. Accepted Ap
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
Schnelle Löser für Partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
A New Unconditionally Stable Method for Telegraph Equation Based on Associated Hermite Orthogonal Functions
The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods
A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
We linearize the Einstein-scalar field equations, expressed relative to
constant mean curvature (CMC)-transported spatial coordinates gauge, around
members of the well-known family of Kasner solutions on . The Kasner solutions model a spatially uniform scalar field
evolving in a (typically) spatially anisotropic spacetime that expands towards
the future and that has a "Big Bang" singularity at . We
place initial data for the linearized system along and study the linear solution's behavior in the collapsing
direction . Our first main result is the proof of an
approximate monotonicity identity for the linear solutions. Using it, we
prove a linear stability result that holds when the background Kasner solution
is sufficiently close to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW)
solution. In particular, we show that as , various
time-rescaled components of the linear solution converge to regular functions
defined along . In addition, we motivate the preferred
direction of the approximate monotonicity by showing that the CMC-transported
spatial coordinates gauge can be viewed as a limiting version of a family of
parabolic gauges for the lapse variable; an approximate monotonicity identity
and corresponding linear stability results also hold in the parabolic gauges,
but the corresponding parabolic PDEs are locally well-posed only in the
direction . Finally, based on the linear stability results, we
outline a proof of the following result, whose complete proof will appear
elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing
direction under small perturbations of its data at .Comment: 73 page
"Mariage des Maillages": A new numerical approach for 3D relativistic core collapse simulations
We present a new 3D general relativistic hydrodynamics code for simulations
of stellar core collapse to a neutron star, as well as pulsations and
instabilities of rotating relativistic stars. It uses spectral methods for
solving the metric equations, assuming the conformal flatness approximation for
the three-metric. The matter equations are solved by high-resolution
shock-capturing schemes. We demonstrate that the combination of a finite
difference grid and a spectral grid can be successfully accomplished. This
"Mariage des Maillages" (French for grid wedding) approach results in high
accuracy of the metric solver and allows for fully 3D applications using
computationally affordable resources, and ensures long term numerical stability
of the evolution. We compare our new approach to two other, finite difference
based, methods to solve the metric equations. A variety of tests in 2D and 3D
is presented, involving highly perturbed neutron star spacetimes and
(axisymmetric) stellar core collapse, demonstrating the ability to handle
spacetimes with and without symmetries in strong gravity. These tests are also
employed to assess gravitational waveform extraction, which is based on the
quadrupole formula.Comment: 29 pages, 16 figures; added more information about convergence tests
and grid setu
Gravitational waves in dynamical spacetimes with matter content in the Fully Constrained Formulation
The Fully Constrained Formulation (FCF) of General Relativity is a novel
framework introduced as an alternative to the hyperbolic formulations
traditionally used in numerical relativity. The FCF equations form a hybrid
elliptic-hyperbolic system of equations including explicitly the constraints.
We present an implicit-explicit numerical algorithm to solve the hyperbolic
part, whereas the elliptic sector shares the form and properties with the well
known Conformally Flat Condition (CFC) approximation. We show the stability
andconvergence properties of the numerical scheme with numerical simulations of
vacuum solutions. We have performed the first numerical evolutions of the
coupled system of hydrodynamics and Einstein equations within FCF. As a proof
of principle of the viability of the formalism, we present 2D axisymmetric
simulations of an oscillating neutron star. In order to simplify the analysis
we have neglected the back-reaction of the gravitational waves into the
dynamics, which is small (<2 %) for the system considered in this work. We use
spherical coordinates grids which are well adapted for simulations of stars and
allow for extended grids that marginally reach the wave zone. We have extracted
the gravitational wave signature and compared to the Newtonian quadrupole and
hexadecapole formulae. Both extraction methods show agreement within the
numerical errors and the approximations used (~30 %).Comment: 17 pages, 9 figures, 2 tables, accepted for publication in PR
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