xx IDBrillData

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 where is a free function subject to certain regularity and fall-off conditions, and is a conformal factor to be solved for.

Thorn IDBrillData provides three choices for the function: an exponential form, (IDBrillData::qxfunction

= "exp")

a generalized form of the function first written down by Eppley (IDBrillData::qxfunction = "eppley") and the (default) Gundlach function which includes the Holz form (IDBrillData::qxfunction = "gundlach")

Substituting the metric into the Hamiltonian constraint gives an elliptic equation for the conformal factor which is then numerically solved for a given function : where the conformal Ricci scalar is found to be

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

In the case of axisymmetry (that is in the above expressions for ), the Hamiltonian constraint can be written as an elliptic equation for with just the flat space Laplacian, If the initial data is chosen to be ADMBase::initialxdata = "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 initialxdata to be brilldata, or for the case of axisymmetry brilldata2D can also be used.

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

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

     Exponential : IDBrillData::qxfunction = "exp"
     (expxa, expxb, expxsigmaz,expxrho0)

     Eppley : IDBrillData::qxfunction = "eppley"
     (eppleyxa, eppleyxb, eppleyxsigmarho, eppleyxr0, eppleyxsigmar, eppleyxc)

     Gundlach : IDBrillData::qxfunction = "gundlach"
              (gundlachxa, gundlachxb, gundlachxsigmarho, gundlachxr0, gundlachxsigmar,
     gundlachxc)

Non-axisymmetric part for each choice of

     (brill3dxd, brill3dxm, brill3dxe, brill3dxn, brill3dxphi0)

Note that the default expression is

.  q = gundlach a ρ^{2}e^{−r^{2}}

IDBrillData can use the elliptic solvers (type LinMetric) provided by CactusEinstein/EllSOR, AEIThorns/BAMxElliptic, 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 brillxsolver:

sor: Understands the Robin boundary condition, additional parameters control the maximum number of iterations (sorxmaxit).

bam: BAMxElliptic does not properly implement the elliptic infrastructure of EllBase, and the BAMxElliptic parameter to use the Robin boundary condition must be set independently of

     IDBrillWave::brillxbound.

3 Notes

Thorn IDBrillData understands both the “physical” and “static conformal” metricxtype. In the case of a

conformal metric being chosen, the conformal factor is set to . Currently the derivatives of the conformal

factor are not calculated, so that only staticconformal::conformalxstorage = "factor" is

supported.

4 References

4.1 Specification of Brill Waves

Dieter Brill, Ann. Phys., 7, 466, 1959.

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 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

11bound 10Scope: privatex01 KEYWORD

11Description: Which boundary condition to use

11 Range 10 x01 Default: robin
11const  10constant boundary: set constxv0
11robin  10Robin boundary: set robinxfalloff, robinxinf

11

11brill3dxd  10Scope: privatex01  REAL

11Description: 3D Brill wave: d rhom̂ cosˆ2 (n (phi + phi0))

11 Range 10 x01 Default: 0.0
11:  10Anything

11

11brill3dxe  10Scope: privatex01  REAL

11Description: 3D Brill wave: d rhom̂ cosˆ2 (n (phi + phi0))

11 Range 10 x01 Default: 1.0
11:  10Anything

11

11brill3dxm  10Scope: privatex01  REAL

11Description: 3D Brill wave: d rhom̂ cosˆ2 (n (phi + phi0))

11 Range 10 x01 Default: 2.0
11:  10Anything

11

11brill3dxn  10Scope: privatex01  REAL

11Description: 3D Brill wave: d rhom̂ cosˆ2 (n (phi + phi0))

11 Range 10 x01 Default: 2.0
11:  10Anything

11

11brill3dxphi0  10Scope: privatex01  REAL

11Description: 3D Brill wave: d rhom̂ cosˆ2 (n (phi + phi0))

11 Range 10 x01 Default: 0.0
11:  10Anything

11

11constxv0  10Scope: privatex01  REAL

11Description: Value of constant BC

11 Range 10 x01 Default: 1.0
11:  10anything goes

11eppleyxa  10Scope: privatex01  REAL

11Description: Eppley Brill wave: Amplitude

11 Range 10 x01 Default: 0.0
11:  10Anything

11

11eppleyxb  10Scope: privatex01  INT

11Description: Eppley Brill wave: used in exponent in rho: rhob̂

11 Range 10 x01 Default: 2
11:  10Anything

11

11eppleyxc  10Scope: privatex01  INT

11Description: Eppley Brill wave: (rˆ2 - r0ˆ2 )ˆ( c/2)

11 Range 10 x01 Default: 2
11:  10Anything

11

11eppleyxr0  10Scope: privatex01  REAL

11Description: Eppley Brill wave: radius of torus in r

11 Range 10 x01 Default: 0.0
110:*  10Positive

11

11eppleyxsigmar 10Scope: privatex01  REAL

11Description: Eppley Brill wave: sigma in r

11 Range 10 x01 Default: 1.0
11:  10Anything

11

11eppleyxsigmarho 10Scope: privatex01  REAL

11Description: Eppley Brill wave: sigma in rho

11 Range 10 x01 Default: 1.0
11:  10Anything

11

11expxa  10Scope: privatex01  REAL

11Description: Exp Brill wave: Amplitude

11 Range 10 x01 Default: 0.0
11:  10Anything

11

11expxb  10Scope: privatex01  INT

11Description: Exp Brill wave: used in exponent in rho: rhoˆ( 2+b)

11 Range 10 x01 Default: 2
11:  10Anything

11

11expxrho0  10Scope: privatex01  REAL

11Description: Exp Brill wave: radius of torus in rho

11 Range 10 x01 Default: 0.0
110:*  10Positive

11

11expxsigmaz  10Scope: privatex01  REAL

11Description: Exp Brill wave: sigma in z

11 Range 10 x01 Default: 1.0
11(0:*  10Positive

11

11gundlachxa  10Scope: privatex01  REAL

11Description: Gundlach Brill wave: Amplitude

11 Range 10 x01 Default: 0.0
11:  10Anything

11gundlachxb  10Scope: privatex01  INT

11Description: Gundlach Brill wave: used in exponent in rho: rhob̂

11 Range 10 x01 Default: 2
11:  10Anything

11

11gundlachxc  10Scope: privatex01  INT

11Description: Gundlach Brill wave: (rˆ2 - r0ˆ2 )ˆ( c/2)

11 Range 10 x01 Default: 2
11:  10Anything

11

11gundlachxr0  10Scope: privatex01  REAL

11Description: Gundlach Brill wave: radius of torus in r

11 Range 10 x01 Default: 0.0
110:*  10Positive

11

11gundlachxsigmar10Scope: privatex01  REAL

11Description: Gundlach Brill wave: sigma in r

11 Range 10 x01 Default: 1.0
11:  10Anything

11

11gundlachxsigmarho 10Scope: privatex01  REAL

11Description: Gundlach Brill wave: sigma in rho

11 Range 10 x01 Default: 1.0
11:  10Anything

11

11outputxcoeffs 10Scope: privatex01  BOOLEAN

11Description: output coefficients for elliptic solve

11               10  x01  Default: no

11

11qxfunction  10Scope: privatex01  KEYWORD

11Description: Form of function q [0,1,2]

11 Range 10 x01 Default: gundlach
11exp  10contains e ˆ{ -zˆ2 } factor
11eppley  10contains 1/(1+rˆ2 ) factor
11gundlach  10contains e ˆ{ -rˆ2 } factor

11

11rhofudge  10Scope: privatex01  REAL

11Description: delta rho for axis fudge

11 Range 10 x01 Default: 0.00001
110:  10Positive please

11

11robinxfalloff  10Scope: privatex01  INT

11Description: Fall-off of Robin BC

11 Range 10 x01 Default: 1
110:  10any positive integer value

11

11robinxinf  10Scope: privatex01  REAL

11Description: Value at infinity of Robin BC

11 Range 10 x01 Default: 1.0
11:  10anything goes

11solver  10Scope: privatex01  KEYWORD

11Description: Which elliptic solver to use

11 Range 10 x01 Default: sor
11sor  10Use SOR solver
11petsc  10Use PETSc solver
11bam  10Use bam solver

11

11sorxmaxit  10Scope: privatex01  INT

11Description: Maximum number of iterations

11 Range 10 x01 Default: 10000
110:*  10Positive

11

11thresh  10Scope: privatex01  REAL

11Description: How far (absolute norm) to go

11 Range 10 x01 Default: 0.00001
110.0:  10Positive number please

11

11conformalxstorage 10Scope: shared from STATICCONFORMALx01KEYWORD

11

6 Interfaces

General Implements: idbrilldata

Inherits: grid admbase staticconformal ellbase

Grid Variables

6.0.1 PRIVATE GROUPS

11x Group Names x10x Variable Names xx01 Details x11x
11  brillelliptic  10  x01  compact110
11                 10  brillMlinear  x01 description11Coefficients for linear elliptic equation
11                 10  brillNsource  x01 dimensions113
11                 10  x01 distribution11DEFAULT
11                 10  x01 group type11GF
11                 10  x01 timelevels111
11                 10  x01variable type11REAL
11  brillconf  10  x01  compact110
11                 10  brillpsi  x01 description11Conformal factor for Brill data
11                 10  x01 dimensions113
11                 10  x01 distribution11DEFAULT
11                 10  x01 group type11GF
11                 10  x01 timelevels111
11                 10  x01variable type11REAL

11

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 x

11x11Conditional:
11x 11brillconf
11x 11brillelliptic

11x 11x

11

Scheduled Functions

CCTKxPARAMCHECK (conditional) x idbrilldataxparamchecker x check that the metricxtype is recognised x

11x11Language:11c
11 x 11Options: 11global
11 x 11Type: 11function

11

CCTKxWRAGH (conditional) x brilldataxinitsymbound x set up symmetries for brillpsi x

11x11Language:11c
11 x 11Options: 11global
11 x 11Type: 11function

11

ADMBasexInitialData (conditional) x brilldata x construct brill wave initial data x

11x11Language:11fortran
11 x 11Type: 11function