IDBrillData

Carsten Gundlach, Gabrielle Allen

Date

Abstract

This thorn creates time symmetric initial data for Brill wave spacetimes. It can create both axisymmetric data (in a 3D cartesian grid), as well as data with an angular dependency.

1 Purpose

The purpose of this thorn is to create (time symmetric) initial data for a Brill wave spacetime. It does so by starting from a three–metric of the form originally considered by Brill

ds2 = Ψ4 e2q dρ2 + dz2 + ρ2dϕ2 = Ψ4dŝ2, (1)

where q is a free function subject to certain regularity and fall-off conditions, ρ = x2 + y2 and Ψ is a conformal factor to be solved for.

Thorn IDBrillData provides three choices for the q function: an exponential form, (IDBrillData::q_function = "exp")

q = aρ2+b r2 e z σz2 e(ρρ0)2 1 + d ρm 1 + eρm cos2 nϕ + ϕ 0 (2)

a generalized form of the q function first written down by Eppley (IDBrillData::q_function = "eppley")

q = a ρ σρ b 1 1 + r2 r02 σr2 c2 1 + d ρm 1 + eρm cos2 nϕ + ϕ 0 (3)

and the (default) Gundlach q function which includes the Holz form (IDBrillData::q_function = "gundlach")

q = a ρ σρ ber2r 02 σ r2 c2 1 + d ρm 1 + eρm cos2 nϕ + ϕ 0 (4)

Substituting the metric into the Hamiltonian constraint gives an elliptic equation for the conformal factor Ψ which is then numerically solved for a given function q:

̂Ψ Ψ 8 R̂ = 0 (5)

where the conformal Ricci scalar is found to be

R̂ = 2 e2q( z2q + ρ2q) + 1 ρ2(3(ϕq)2 + 2 ϕq) (6)

Assuming the initial data to be time symmetric means that the momentum constraints are trivially satisfied.

In the case of axisymmetry (that is d = 0 in the above expressions for q), the Hamiltonian constraint can be written as an elliptic equation for Ψ with just the flat space Laplacian,

flatΨ + Ψ 4 (z2q + ρ2q) = 0 (7)

If the initial data is chosen to be ADMBase::initial_data = "brilldata2D" then this elliptic equation is solved rather than the equation above.

2 Generating Initial Data with IDBrillData

Brill initial data is activated by choosing the CactusEinstein/ADMBase parameter initial_data to be brilldata, or for the case of axisymmetry brilldata2D can also be used.

The parameter IDBrillData::q_function chooses the form of the q function to be used, defaulting to the Gundlach expression.

Additional IDBrillData parameters for each form of q fix the remaining freedom:

Note that the default q expression is

q = gundlach_aρ2er2

IDBrillData can use the elliptic solvers (type LinMetric) provided by CactusEinstein/EllSOR,
AEIThorns/BAM_Elliptic, or CactusElliptic/EllPETSc to solve the equation resulting from the Hamiltonian constraint. In all cases the parameter thresh sets the threshold for the elliptic solve. The choice of elliptic solver is made through the parameter brill_solver:

3 Notes

Thorn IDBrillData understands both the “physical” and “static conformalmetric_type. In the case of a conformal metric being chosen, the conformal factor is set to Ψ. Currently the derivatives of the conformal factor are not calculated, so that only staticconformal::conformal_storage = "factor" is supported.

4 References

4.1 Specification of Brill Waves

  1. Dieter Brill, Ann. Phys., 7, 466, 1959.
  2. Ken Eppley, Sources of Gravitational Radiation, edited by L. Smarr (Cambridge University Press, Cambridge, England, 1979), p. 275.

4.2 Numerical Evolutions of Brill Waves

  1. Gravitational Collapse of Gravitational Waves in 3D Numerical Relativity, Miguel Alcubierre, Gabrielle Allen, Bernd Bruegmann, Gerd Lanfermann, Edward Seidel, Wai-Mo Suen, Malcolm Tobias, Phys. Rev. D61, 041501, 2000.

5 Parameters




bound
Scope: private  KEYWORD



Description: Which boundary condition to use



Range   Default: robin
const
constant boundary: set const_v0
robin
Robin boundary: set robin_falloff, robin_inf






brill3d_d
Scope: private  REAL



Description: 3D Brill wave: d rhom̂ cos2̂(n (phi + phi0))



Range   Default: 0.0
:
Anything






brill3d_e
Scope: private  REAL



Description: 3D Brill wave: d rhom̂ cos2̂(n (phi + phi0))



Range   Default: 1.0
:
Anything






brill3d_m
Scope: private  REAL



Description: 3D Brill wave: d rhom̂ cos2̂(n (phi + phi0))



Range   Default: 2.0
:
Anything






brill3d_n
Scope: private  REAL



Description: 3D Brill wave: d rhom̂ cos2̂(n (phi + phi0))



Range   Default: 2.0
:
Anything






brill3d_phi0
Scope: private  REAL



Description: 3D Brill wave: d rhom̂ cos2̂(n (phi + phi0))



Range   Default: 0.0
:
Anything






const_v0
Scope: private  REAL



Description: Value of constant BC



Range   Default: 1.0
:
anything goes






eppley_a
Scope: private  REAL



Description: Eppley Brill wave: Amplitude



Range   Default: 0.0
:
Anything






eppley_b
Scope: private  INT



Description: Eppley Brill wave: used in exponent in rho: rhob̂



Range   Default: 2
:
Anything






eppley_c
Scope: private  INT



Description: Eppley Brill wave: (r2̂ - r02̂)(̂c/2)



Range   Default: 2
:
Anything






eppley_r0
Scope: private  REAL



Description: Eppley Brill wave: radius of torus in r



Range   Default: 0.0
0:*
Positive






eppley_sigmar
Scope: private  REAL



Description: Eppley Brill wave: sigma in r



Range   Default: 1.0
:
Anything






eppley_sigmarho
Scope: private  REAL



Description: Eppley Brill wave: sigma in rho



Range   Default: 1.0
:
Anything






exp_a
Scope: private  REAL



Description: Exp Brill wave: Amplitude



Range   Default: 0.0
:
Anything






exp_b
Scope: private  INT



Description: Exp Brill wave: used in exponent in rho: rho(̂2+b)



Range   Default: 2
:
Anything






exp_rho0
Scope: private  REAL



Description: Exp Brill wave: radius of torus in rho



Range   Default: 0.0
0:*
Positive






exp_sigmaz
Scope: private  REAL



Description: Exp Brill wave: sigma in z



Range   Default: 1.0
(0:*
Positive






gundlach_a
Scope: private  REAL



Description: Gundlach Brill wave: Amplitude



Range   Default: 0.0
:
Anything






gundlach_b
Scope: private  INT



Description: Gundlach Brill wave: used in exponent in rho: rhob̂



Range   Default: 2
:
Anything






gundlach_c
Scope: private  INT



Description: Gundlach Brill wave: (r2̂ - r02̂)(̂c/2)



Range   Default: 2
:
Anything






gundlach_r0
Scope: private  REAL



Description: Gundlach Brill wave: radius of torus in r



Range   Default: 0.0
0:*
Positive






gundlach_sigmar
Scope: private  REAL



Description: Gundlach Brill wave: sigma in r



Range   Default: 1.0
:
Anything






gundlach_sigmarho
Scope: private  REAL



Description: Gundlach Brill wave: sigma in rho



Range   Default: 1.0
:
Anything






output_coeffs
Scope: private  BOOLEAN



Description: output coefficients for elliptic solve



  Default: no






q_function
Scope: private  KEYWORD



Description: Form of function q [0,1,2]



Range   Default: gundlach
exp
contains e{̂-z2̂} factor
eppley
contains 1/(1+r2̂) factor
gundlach
contains e{̂-r2̂} factor






rhofudge
Scope: private  REAL



Description: delta rho for axis fudge



Range   Default: 0.00001
0:
Positive please






robin_falloff
Scope: private  INT



Description: Fall-off of Robin BC



Range   Default: 1
0:
any positive integer value






robin_inf
Scope: private  REAL



Description: Value at infinity of Robin BC



Range   Default: 1.0
:
anything goes






solver
Scope: private  KEYWORD



Description: Which elliptic solver to use



Range   Default: sor
sor
Use SOR solver
petsc
Use PETSc solver
bam
Use bam solver






sor_maxit
Scope: private  INT



Description: Maximum number of iterations



Range   Default: 10000
0:*
Positive






thresh
Scope: private  REAL



Description: How far (absolute norm) to go



Range   Default: 0.00001
0.0:
Positive number please






conformal_storage
Scope: shared from STATICCONFORMAL KEYWORD



6 Interfaces

General

Implements:

idbrilldata

Inherits:

grid

admbase

staticconformal

ellbase

Grid Variables

6.0.1 PRIVATE GROUPS




  Group Names    Variable Names    Details   




brillelliptic   compact0
brillMlinear   descriptionCoefficients for linear elliptic equation
brillNsource   dimensions3
  distributionDEFAULT
  group typeGF
  timelevels1
 variable typeREAL




brillconf   compact0
brillpsi   descriptionConformal factor for Brill data
  dimensions3
  distributionDEFAULT
  group typeGF
  timelevels1
 variable typeREAL




7 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/IDBrillData. 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

 

 Conditional:
  brillconf
  brillelliptic
   

Scheduled Functions

CCTK_PARAMCHECK (conditional)

  idbrilldata_paramchecker

  check that the metric_type is recognised

 

 Language:c
 Options: global
 Type: function

CCTK_WRAGH (conditional)

  brilldata_initsymbound

  set up symmetries for brillpsi

 

 Language:c
 Options: global
 Type: function

ADMBase_InitialData (conditional)

  brilldata

  construct brill wave initial data

 

 Language:fortran
 Type: function