After a formalism for evaluating QE irrotational NS-NS sequences was developed [51, 313
], some of the
first results were obtained by Uryƫ and Eriguchi, who developed a finite-differencing code in spherical
coordinates allowing for the solution of relativistic NS-NS binaries using Green’s functions[313
, 317
].
Their method extended the self-consistent field (SCF) work of [147], which had previously
been applied to axisymmetric configurations. Irrotational configurations were also generated by
Marronetti et al. [185], using the same finite difference scheme as found in the work on synchronized
binaries.
The most widely used direct grid-based solver in numerical relativity is the Bam_Elliptic solver [55],
which solves elliptic equations on single rectangular grids or multigrid configurations. It is included within
the Cactus code, which is widely used in 3-D numerical relativity [307]. In particular it has been used to
initiate a number of single and binary BH simulations, including one of the original breakthrough binary
puncture works [61].
Lagrangian methods, typically based on smoothed particle hydrodynamics (SPH) [181, 118, 194]) have
been used to generate both synchronized and irrotational configurations for PN [10, 99
, 101
, 100
] and
conformally flat (CF) [211
, 210
, 97
, 212
, 207
, 209
, 208
, 34
, 35
, 33
] calculations of NS-NS mergers, but
they have not yet been extended to fully GR calculations, in part because of the difficulties in evolving the
global spacetime metric.
The most widely used data for numerical calculations are those generated by the Meudon group (see
Section 4.4 below for details on their calculations and [125] for a detailed description of their methods).
The code they developed, Lorene [124
], uses multidomain spectral methods to solve elliptic equations
(while the code has been used primarily for relativistic stellar and binary configurations, it can be used as a
more general solver). Around each star, one creates a set of nested, contiguous grids, with points arrayed in
the radial and angular directions. The innermost grid has spheroidal geometry, and the surrounding grids
are annular. The outermost grid may be allowed to extend to spatial infinity through a compactification
transformation of the radial coordinate. To solve elliptic equations for various field quantities, one breaks
each into a sum of two components, each of whose source terms are concentrated in one NS
or the other. Similarly, the source terms themselves are split into two pieces, ideally, so each
component is well-described by spheroidal spectral coefficients centered around each star. Using the
spectral expansion, one may pass values from one star to the other and then recalculate spectral
coefficients for the other grid configuration. This scheme has several efficiency advantages over direct
grid-based methods, which helps to explain its popularity. First, the domain geometry may be
chosen to fit to a NS surface, which eliminates Gibbs phenomenon-related errors and allows for
exponential convergence with respect to the number of grid points, rather than the geometric
convergence that characterizes finite difference-based grid codes. Second, the use of spectral
methods requires much less computer memory than grid-based codes, and, as a result, Lorene is a
serial code that can run easily on any off-the-shelf PC, rather than requiring a supercomputer
platform.
http://www.livingreviews.org/lrr-2012-8 |
Living Rev. Relativity 15, (2012), 8
![]() This work is licensed under a Creative Commons License. E-mail us: |