1 Volume integrals

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

1.1 Rest mass

We start by defining the “densitised density” (see e.g. [1]) \begin {equation} \rho _{\star } \equiv \alpha \sqrt {\gamma }\rho _{\rm b} u^{0} = W\rho _{\rm b}\sqrt {\gamma }, \end {equation} where \(\alpha \) is the lapse function, \(\gamma \) is the determinant of the physical spatial metric \(\gamma _{ij}\), \(\rho _{\rm b}\) is the baryonic density, \(u^{\mu }\) is the fluid four-velocity, and \(W\equiv \alpha u^{0}\) is the Lorentz factor. The rest mass integral is then given by \begin {equation} \boxed { M_{0} = \int \rho _{\star } dV = \int \left (W\rho _{\rm b}\sqrt {\gamma }\right )dV }\; . \end {equation} The volume \(V\) can be specified using spherical shells, regions with hollowed balls, and simple spheres. It can also be set to the entire computational domain.

1.2 Center of mass

The coordinates of center of mass \(X^{i}\) are computed from the integral \begin {equation} \boxed { X^{i} = \frac {\int \rho _{\star } x^{i}dV}{\int \rho _{\star } dV} }\; , \end {equation} where \(x^{i}\) is the position vector. The volume \(V\) can be specified using spherical shells, regions with hollowed balls, and simple spheres. It can also be set to the entire domain size.

1.3 Unit (for debugging)

This thorn also provides the functionality of performing a volume integral with a unit integrand, which should just yield the volume of the integrated region, i.e. \begin {equation} \boxed { V = \int dV }\; . \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:

1. Integrate the entire volume with an integrand of 1. (used for testing/validation purposes only).

2. Perform all four integrals required to compute the center of mass (numerator=3, denominator=1 integral), in an integration volume originally centered at \((x,y,z)=(-15.2,0,0)\) with a coordinate radius of 13.5 Also use the center of mass integral result to SET the ZEROTH AMR center. In this way, the AMR grids will track the center of mass as computed by this integration.

3. Same as 2, except use the integrand=1 (for validation purposes, to ensure the integration volume is approximately \((4/3)\pi 13.5^3\)). Also disable tracking of this integration volume.

4. Same as 2, except for the neutron star originally centered at \((x,y,z)=(+15.2,0,0)\).

5. Same as 4, except use the integrand=1 (for validation purposes, to ensure the integration volume is approximately \((4/3)\pi 13.5^3\)). Also disable tracking of this integration volume.

6. Perform rest-mass integrals over entire volume.

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

ActiveThorns = "VolumeIntegrals_GRMHD"

# Set number of integrals
VolumeIntegrals_GRMHD::NumIntegrals = 6

# Set frequency of integration. If tracking AMR centres this should
# match the regridding frequency of the thorn CactusRegrid2
VolumeIntegrals_GRMHD::VolIntegral_out_every = 32

# Enable file output
VolumeIntegrals_GRMHD::enable_file_output = 1

# Set output directory
VolumeIntegrals_GRMHD::outVolIntegral_dir = "volume_integration"

# Set verbose level. 0 disables it, 1 provides useful
# information, and 2 is very verbose.
VolumeIntegrals_GRMHD::verbose = 1

# The AMR centre will only track the first referenced
# integration quantities that track said centre.

# Integral #1: unit integrand
VolumeIntegrals_GRMHD::Integration_quantity_keyword[1] = "one"
# Integral #2: CoM
VolumeIntegrals_GRMHD::Integration_quantity_keyword[2] = "centerofmass"
# Integral #3: unit integrand
VolumeIntegrals_GRMHD::Integration_quantity_keyword[3] = "one"
# Integral #4: CoM
VolumeIntegrals_GRMHD::Integration_quantity_keyword[4] = "centerofmass"
# Integral #5: unit integrand
VolumeIntegrals_GRMHD::Integration_quantity_keyword[5] = "one"
# Integral #6: rest mass
VolumeIntegrals_GRMHD::Integration_quantity_keyword[6] = "restmass"

# Set radius of integration for integral #2
VolumeIntegrals_GRMHD::volintegral_sphere__center_x_initial         [2] = -15.2
VolumeIntegrals_GRMHD::volintegral_inside_sphere__radius            [2] =  13.5
# Let the result of integral #2 track the grid centre 0
VolumeIntegrals_GRMHD::amr_centre__tracks__volintegral_inside_sphere[2] =  0

# Set radius of integrattion for integral #3
VolumeIntegrals_GRMHD::volintegral_sphere__center_x_initial         [3] = -15.2
VolumeIntegrals_GRMHD::volintegral_inside_sphere__radius            [3] =  13.5

                                                                                       
                                                                                       
# Set radius of integration for integral #4
VolumeIntegrals_GRMHD::volintegral_sphere__center_x_initial         [4] =  15.2
VolumeIntegrals_GRMHD::volintegral_inside_sphere__radius            [4] =  13.5
# Let the result of integral #4 track the grid centre 1
VolumeIntegrals_GRMHD::amr_centre__tracks__volintegral_inside_sphere[4] =  1

# Set radius of integrattion for integral #5
VolumeIntegrals_GRMHD::volintegral_sphere__center_x_initial         [5] =  15.2
VolumeIntegrals_GRMHD::volintegral_inside_sphere__radius            [5] =  13.5

References

3 Parameters

[1]

usepreviousintegrands

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_grmhd

Inherits:

grid

admbase

carpetregrid2

hydrobase

admbase

Grid Variables

4.0.1 PRIVATE GROUPS

5 Schedule

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

  create directory for vi grmhd file output.

CCTK_POST_RECOVER_VARIABLES (conditional)

  vi_grmhd_file_output_routine_startup

  create directory for vi grmhd file output.

CCTK_INITIAL

  vi_grmhd_initializeintegralcountertozero

  initialize integralcounter variable to zero

CCTK_POST_RECOVER_VARIABLES

  vi_grmhd_initializeintegralcountertozero

  initialize integralcounter variable to zero

CCTK_ANALYSIS

  vi_grmhd_initializeintegralcounter

  initialize integralcounter variable

CCTK_ANALYSIS

  vi_grmhd_volumeintegralgroup

  evaluate all volume integrals

VI_GRMHD_VolumeIntegralGroup

  vi_grmhd_computeintegrand

  compute integrand

VI_GRMHD_VolumeIntegralGroup

  vi_grmhd_dosum

  do sum

VI_GRMHD_VolumeIntegralGroup

  vi_grmhd_decrementintegralcounter

  decrement integralcounter variable

CCTK_ANALYSIS

  vi_grmhd_file_output

  output volumeintegral results to disk