1 Volume integrals

We now briefly describe the volume integrals that can be performed using the VolumeIntegrals_vacuum thorn.

1.1 \(L^{2}\)-norm

For a given field \(f\), the \(L^{2}\)-norm of the field, \(\left \lVert f \right \rVert _{2}\), is computed using the volume integral \begin {equation} \boxed {\left \lVert \right \rVert _{2} = \int f^{2}dV}\; , \end {equation} where \(dV\) is the volume element.

1.2 Center of the lapse

The center of the lapse, \(C_{\alpha }\), volume integral yields results which are pretty consistent with the center of mass volume integral. We compute it using \begin {equation} \boxed { C_{\alpha }^{i} = \int \left [\left (1-\alpha \right )^{80}x^{i}\right ]dV }\; , \end {equation} where \(\alpha \) is the lapse function, \(x^{i}\) is the position vector and \(dV\) is the volume element.

1.3 ADM mass

The ADM mass, \(M_{\rm ADM}\) is computed using equation (A.5) in [1] (see also eq. (7.15) in [2]) \begin {equation} M_{\rm ADM} = \frac {1}{16\pi }\lim _{r\to \infty }\oint _{S}\left [ \delta ^{ij}\left (\partial _{i}h_{jk} - \partial _{k}h_{ij}\right )dS^{k} \right ], \end {equation} where \(S\) is the surface of integration and \(dS^{i} = s^{i}dA\), with \(s^{i}\) the unit outward-pointing normal vector to the surface and \(dA\) the area element. To obtain the equation above, the physical spatial metric, \(\gamma _{ij}\), has been decomposed using \begin {equation} \gamma _{ij} = \delta _{ij} + h_{ij}, \end {equation} where \(\delta _{ij}\) is the Kronecker delta and represents the flat space metric in Cartesian coordinates, while \(h_{ij}\) is a small perturbation around flat space physical spatial metric. In practice, we do not take the integration surface to be at infinity, and therefore we implement the expression \begin {equation} \boxed { M_{\rm ADM} = \frac {1}{16\pi }\oint _{S}\left [ \gamma ^{ij}\left (\partial _{i}\gamma _{jk} - \partial _{k}\gamma _{ij}\right )dS^{k} \right ] }\; . \end {equation} where \(\gamma ^{ij}\) is the inverse spatial metric.

1.4 ADM momentum

The ADM momentum, \(P_{\rm ADM}^{i}\), is obtained from equation (A.6) in [1] (see also eq. (7.56) in [2]) \begin {equation} P_{\rm ADM}^{i} = \frac {1}{8\pi }\lim _{r\to \infty }\oint _{S}\left [ \left (K^{i}_{\ j}-\delta ^{i}_{\ j}K\right )dS^{j} \right ], \end {equation} where \(K_{ij}\) is the extrinsic curvature and \(K\equiv \gamma ^{ij}K_{ij}\) is the mean curvature. Like the ADM mass, the integration is not performed at infinity, and the implemented equation is \begin {equation} \boxed { P_{\rm ADM}^{i} = \frac {1}{8\pi }\oint _{S}\left [ \left (K^{ij}-\gamma ^{ij}K\right )dS_{j} \right ] }\; . \end {equation}

1.5 ADM angular momentum

The ADM angular momentum, \(J_{\rm ADM}^{i}\), follows from equation (A.7) in [1] (see also eq. (7.63) in [2]) \begin {equation} J_{\rm ADM}^{i} = \frac {1}{8\pi }\lim _{r\to \infty }\oint _{S}\left [ \epsilon ^{ijk}x_{j}\left (K_{kl} - \delta _{kl}K\right )dS^{l} \right ], \end {equation} where \(\epsilon ^{ijk}\) is the three-dimensional Levi-Civita tensor and \(x^{i}\) are the components of the position vector in Cartesian coordinates. At a finite distance from the origin this equation is written as \begin {equation} \boxed { J_{\rm ADM}^{i} = \frac {1}{8\pi }\oint _{S}\left [ \epsilon ^{ijk}x_{j}\left (K_{kl} - \gamma _{kl}K\right )dS^{l} \right ] }\; . \end {equation}

2 Basic usage

Except for the definition of the integrands, the behavior of this thorn is almost completely driver by the configuration of the parameter file. We present here an example of such a parameter file, which performs the following tasks:

2.1 Specifying the BSSN evolution thorn

The VolumeIntegrals_vacuum thorn requires information about the Hamiltonian and momentum constraint variables in order to perform certain volume integrals. For maximum flexibility, one can specify which variables to use, making VolumeIntegrals_vacuum compatible with any BSSN evolution thorn. This is achieved by setting the following variables:

1. VolumeIntegrals_vacuum::HamiltonianVarString;

2. VolumeIntegrals_vacuum::Momentum0VarString;

3. VolumeIntegrals_vacuum::Momentum1VarString;

4. VolumeIntegrals_vacuum::Momentum2VarString;

The default values for these variables are the variables from the ML_BSSN thorn, but you can use any thorn you want. For example, to use the variables Ham, MU0, MU1, and MU2 from an evolution thorn called MyBSSNthorn, one would add the following lines to the parameter file:

VolumeIntegrals_vacuum::HamiltonianVarString = "MyBSSNthorn::Ham"
VolumeIntegrals_vacuum::Momentum0VarString   = "MyBSSNthorn::MU0"
VolumeIntegrals_vacuum::Momentum1VarString   = "MyBSSNthorn::MU1"
VolumeIntegrals_vacuum::Momentum2VarString   = "MyBSSNthorn::MU2"

2.2 Full parameter file configuration example

We now provide an example of a parameter file configuration which uses the Hamiltonian and momentum constraint variables of the Baikal thorn and performs the following tasks:

1. Integral of Hamiltonian & momentum constraints, over the entire grid volume.

2. Exactly the same as 1, but excising the region inside a sphere of radius 2.2 tracking the zeroth AMR grid (initially at \((x,y,z) = (-5.9,0,0)\));

3. Exactly the same as 2, but additionally excising the region inside a sphere of radius 2.2 tracking the center of the first AMR grid (initially at \((x,y,z)=(+5.9,0,0)\));

4. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 8.2;

5. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 12.0;

6. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 16.0;

7. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 24.0;

8. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 48.0;

9. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 96.0;

10. Integral of Hamiltonian & momentum constraints, over the entire grid volume, minus the spherical region inside coordinate radius 192.0;

To achieve this, add the following configuration to your parameter file:

ActiveThorns = "VolumeIntegrals_vacuum"
# Set the Hamiltonian and momentum constraint variables to Baikal’s
VolumeIntegrals_vacuum::HamiltonianVarString = "Baikal::HGF"
VolumeIntegrals_vacuum::Momentum0VarString   = "Baikal::MU0GF"
VolumeIntegrals_vacuum::Momentum1VarString   = "Baikal::MU1GF"
VolumeIntegrals_vacuum::Momentum2VarString   = "Baikal::MU2GF"

# Now setup basic information about the integrals
VolumeIntegrals_vacuum::NumIntegrals = 10
VolumeIntegrals_vacuum::VolIntegral_out_every = 64
VolumeIntegrals_vacuum::enable_file_output = 1
VolumeIntegrals_vacuum::outVolIntegral_dir = "volume_integration"
VolumeIntegrals_vacuum::verbose = 1
# The AMR centre will only track the first referenced integration
# quantities that track said centre. Thus, centeroflapse output will
# not feed back into the AMR centre positions.
VolumeIntegrals_vacuum::Integration_quantity_keyword[1] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[2] = "usepreviousintegrands"
VolumeIntegrals_vacuum::Integration_quantity_keyword[3] = "usepreviousintegrands"
VolumeIntegrals_vacuum::Integration_quantity_keyword[4] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[5] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[6] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[7] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[8] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[9] = "H_M_CnstraintsL2"
VolumeIntegrals_vacuum::Integration_quantity_keyword[10]= "H_M_CnstraintsL2"

# Second integral takes the first integral integrand,
# then excises the region around the first BH
VolumeIntegrals_vacuum::volintegral_sphere__center_x_initial            [2] = -5.9
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius              [2] =  2.2
VolumeIntegrals_vacuum::volintegral_sphere__tracks__amr_centre          [2] =  0
VolumeIntegrals_vacuum::volintegral_usepreviousintegrands_num_integrands[2] =  4

# Third integral takes the second integral integrand,
# then excises the region around the first BH
VolumeIntegrals_vacuum::volintegral_sphere__center_x_initial            [3] =  5.9
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius              [3] =  2.2
VolumeIntegrals_vacuum::volintegral_sphere__tracks__amr_centre          [3] =  1
VolumeIntegrals_vacuum::volintegral_usepreviousintegrands_num_integrands[3] =  4

# Just an outer region
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[4] = 8.2
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[5] =12.0
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[6] =16.0
                                                                                       
                                                                                       
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[7] =24.0
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[8] =48.0
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[9] =96.0
VolumeIntegrals_vacuum::volintegral_outside_sphere__radius[10]=192.0

References

3 Parameters

[1]

usepreviousintegrands

[1]

ADM\_Angular\_Momentum

[1]

BaikalVacuum::MSQUAREDGF

outvolintegral_dir	Scope: private	STRING
Description: Output directory for volume integration output files, overrides IO::out_dir
Range		Default: (none)
.+	A valid directory name
	An empty string to choose the default from IO::out_dir

4 Interfaces

General

Implements:

volumeintegrals_vacuum

Inherits:

grid

admbase

carpetregrid2

Grid Variables

4.0.1 PRIVATE GROUPS

4.0.2 PUBLIC GROUPS

5 Schedule

This section lists all the variables which are assigned storage by thorn WVUThorns_Diagnostics/VolumeIntegrals_vacuum. 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

CCTK_INITIAL (conditional)

  vi_vacuum_file_output_routine_startup

  create directory for file output.

CCTK_POST_RECOVER_VARIABLES (conditional)

  vi_vacuum_file_output_routine_startup

  create directory for file output.

CCTK_INITIAL

  initializeintegralcountertozero

  initialize integralcounter variable to zero

CCTK_POST_RECOVER_VARIABLES

  initializeintegralcountertozero

  initialize integralcounter variable to zero

CCTK_ANALYSIS

  initializeintegralcounter

  initialize integralcounter variable

CCTK_ANALYSIS

  volumeintegralgroup

  evaluate all volume integrals

VolumeIntegralGroup

  volumeintegrals_vacuum_computeintegrand

  compute integrand

VolumeIntegralGroup

  dosum

  do sum

VolumeIntegralGroup

  decrementintegralcounter

  decrement integralcounter variable

CCTK_ANALYSIS

  vi_vacuum_file_output

  output volumeintegral results to disk