1 Introduction

The hyperbolic relaxation method of [1] converts an elliptic problem into a hyperbolic one. The prototypical elliptic problem—Poisson’s equation—reads \begin {equation} \nabla ^{2}\vec {u} = \vec {\rho }\;, \end {equation} where \(\vec {u}\) is a set of unknowns and \(\vec {\rho }\) are sources. One introduces a relaxation time \(t\) and replaces this elliptic equation with \begin {equation} \partial _{t}^{2}\vec {u} + \eta \partial _{t}\vec {u} = c^{2}\left (\nabla ^{2}\vec {u} - \vec {\rho }\right )\;, \end {equation} where \(\eta \) is a damping parameter with units of inverse time and \(c\) is the speed of the relaxation waves. One then evolves this hyperbolic equation until a steady state is reached at \(t=t_{\star }\), for which \begin {equation} \left .\partial _{t}^{2}\vec {u}\right |_{t=t_{\star }} = 0 = \left .\partial _{t}\vec {u}\right |_{t=t_{\star }}\;, \end {equation} and thus \(\left .\vec {u}\right |_{t=t_{\star }}\) is also a solution to the original elliptic problem. For more details of how this method is implemented in NRPyEllipticET, please see [2].

2 Initial data types

2.1 Conformally flat binary black hole initial data

As described in [2], NRPyEllipticET solves the Hamiltonian constraint using the conformal transverse-traceless (CTT) decomposition, i.e., \begin {equation} \hat \nabla ^{2}u + \frac {1}{8}\tilde {A}_{ij}\tilde {A}^{ij}\left (\psi _{\rm singular}+u\right )^{-7} = 0\;, \end {equation} where \(\hat \nabla _{i}\) is the covariant derivative compatible with the flat spatial metric \(\hat \gamma _{ij}\), \(\hat \nabla ^{2} = \hat \nabla _{i}\hat \nabla ^{i}\), \(\tilde {A}_{ij} = \psi ^{-4}A_{ij}\), where \(A_{ij}\) is the traceless part of the extrinsic curvature, and \(\psi \) is the conformal factor, given by \begin {equation} \psi = \psi _{\rm singular} + u \equiv 1 + \sum _{n=1}^{N_{p}}\frac {m_{n}}{2|\vec {x}_{n}|} + u\;, \end {equation} where \(m_{n}\) are \(\vec {x}_{n}\) are the bare mass and position vector of the \(n\)^th puncture and \(u\) is a to-be-determined non-singular piece of the conformal factor.

We thus solve the Hamiltonian constraint using the hyperbolic relaxation method, i.e., \begin {align} \partial _{t}u &= v - \eta u\;,\\ \partial _{t}v &= c^{2}\left [\hat {\nabla }^{2}u + \frac {1}{8}\tilde {A}_{ij}\tilde {A}^{ij}\left (\psi _{\rm singular} + u\right )\right ]\;, \end {align}

where the first equation defines \(v\).

References

3 Parameters

lapse_exponent_n	Scope: private	REAL
Description: Exponent of lapse initial data alpha = psin}
Range		Default: -2
*:*	This parameter can have any value

[1]

NRPyEllipticET-psi\^n

[1]

NRPyEllipticET-averaged

4 Interfaces

General

Implements:

nrpyellipticet

Inherits:

admbase

grid

Grid Variables

4.0.1 PRIVATE GROUPS

5 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/NRPyEllipticET. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

Scheduled Functions

ADMBase_InitialData

  nrpyellipticet

  set up metric fields for binary black hole initial data

ADMBase_InitialData

  nrpyellipticet_cleanup

  deallocate memory for arrays used when computing initial data