4.3 Numerical implementations

There are a number of numerical techniques that have been used to solve these elliptic systems. The first calculations of NS-NS QE sequences, in both cases for synchronized binaries, were performed by Wilson, Mathews, and Marronetti [327Jump To The Next Citation Point, 328Jump To The Next Citation Point, 189Jump To The Next Citation Point] and Baumgarte et al. [25Jump To The Next Citation Point, 26Jump To The Next Citation Point]. The former used a finite differencing scheme, and centered different quantities at cells, vertices, and faces in order to construct a system of equations that was solved using fast matrix inversion techniques, while the latter used a Cartesian multigrid scheme, restricted to an octant to increase computational efficiency.

After a formalism for evaluating QE irrotational NS-NS sequences was developed [51Jump To The Next Citation Point, 313Jump To The Next Citation Point], some of the first results were obtained by UryĆ« and Eriguchi, who developed a finite-differencing code in spherical coordinates allowing for the solution of relativistic NS-NS binaries using Green’s functions[313Jump To The Next Citation Point, 317Jump To The Next Citation Point]. 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 [61Jump To The Next Citation Point].

Lagrangian methods, typically based on smoothed particle hydrodynamics (SPH) [181, 118, 194]) have been used to generate both synchronized and irrotational configurations for PN [10Jump To The Next Citation Point, 99Jump To The Next Citation Point, 101Jump To The Next Citation Point, 100Jump To The Next Citation Point] and conformally flat (CF) [211Jump To The Next Citation Point, 210Jump To The Next Citation Point, 97Jump To The Next Citation Point, 212Jump To The Next Citation Point, 207Jump To The Next Citation Point, 209Jump To The Next Citation Point, 208Jump To The Next Citation Point, 34Jump To The Next Citation Point, 35Jump To The Next Citation Point, 33Jump To The Next Citation Point] 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 [125Jump To The Next Citation Point] for a detailed description of their methods). The code they developed, Lorene [124Jump To The Next Citation Point], 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.

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