Multipole

Ian Hinder

01 Jun 2010

Abstract

The Multipole thorn performs spherical harmonic mode decomposition of Cactus grid functions on coordinate spheres.

1 Introduction

This thorn allows the user to compute the coefficients of the spherical harmonic expansion of a field stored in a Cactus grid function on coordinate spheres of given radii. A set of radii for these spheres, as well as the number of angular points to use, can be specified. Complex fields can be used, but they must be stored in pairs of real Cactus grid functions (the CCTK_COMPLEX type is not supported).

2 Physical System

The angular dependence of a field u(t,r,𝜃,φ) can be expanded in spin-weight s spherical harmonics [1]:

u(t,r,𝜃,φ) = l=0 m=llClm(t,r) sY lm(𝜃,φ) (1)

where the coefficients Clm(t,r) are given by

Clm(t,r) =sY lmu(t,r,𝜃,φ)r2dΩ (2)

At any given time, t, this thorn can compute Clm(t,r) for a number of grid functions on several coordinate spheres with radii ri. The coordinate system of the Cactus grids must be Cartesian and the coordinates r, 𝜃, φ are related to x, y and z by the usual transformation between Cartesian and spherical polar coordinates (𝜃 is the polar angle and φ is the azimuthal angle).

The spin-weighted spherical harmonics are computed using Eq. 3.1 in Ref. [1].

3 Numerical Implementation

The coordinate sphere on which the multipolar decomposition is performed is represented internally as a 2-dimensional grid evenly spaced in 𝜃 and φ with coordinates

𝜃k = πk n𝜃 k = 0,1,...,n𝜃 (3) φk = 2πk nφ k = 0,1,...,nφ, (4)

so n𝜃 and nφ count the number of cells (not the number of points). The gridfunction to be decomposed, u, is first interpolated from the 3D Cactus grid onto this 2D grid at a given radius, ri, and the Clm are computed by evaluating the integral in Eq. 2 for different values of l and m. The interpolation is performed using the Cactus interpolation interface, so any Cactus interpolator can be used. The numerical method used for interpolation should be specified in the documentation for the thorn which provides it. One such thorn is AEILocalInterp. The integration is performed using either the midpoint rule, yielding a result which is first order accurate in the angular spacings Δ𝜃 = πn𝜃 and Δφ = 2πn𝜃, or Simpson’s rule, which is fourth order accurate.

4 Using This Thorn

4.1 Obtaining This Thorn

This thorn is available as part of the Einstein Toolkit via:

svn checkout https://svn.einsteintoolkit.org/cactus/EinsteinAnalysis/Multipole/trunk Multipole

or by using the Einstein Toolkit GetComponents script and thornguide.

4.2 Basic Usage

Suppose that you have a real grid function, u, for which you want to compute the spherical harmonic coefficients Clm. Start by including Multipole and an interpolator (for example AEILocalInterp) in the ActiveThorns line of your parameter file (for interpolation thorns other than AEILocalInterp, you will need to modify the interpolator and interpolator_options accoording to the documentation of the interpolation thorn). Next decide the number and radii of the coordinate spheres on which you want to decompose. Set the number of spheres with the nradii parameter, and the radii themselves with the radius[i] parameters (the indices i are zero-based). For example,

ActiveThorns = ".... AEILocalInterp Multipole"  
 
Multipole::nradii       = 3  
Multipole::radius[0]    = 10  
Multipole::radius[1]    = 20  
Multipole::radius[2]    = 30  
Multipole::variables    = "MyThorn::u"

The default parameters will compute all l = 2 modes assuming a spin-weight s = 0 on every iteration of your simulation. The coefficients Clm will be output in the files with names of the form mp_<var>_l<lmode>_m<mmode>_r<rad>.asc, for example

mp_u_l2_m2_r10.00.asc  
mp_u_l2_m-1_r20.00.asc

For the filename, the radius is converted to a string with two decimal places, which should be sufficient. These are ASCII files where each line has columns

t, ReClm, ImClm.

4.3 Special Behaviour

Often it will be necessary to go beyond the basic usage described in the previous section.

4.3.1 Higher modes

By default, Multipole computes only l = 2 modes. You can choose whether to extract a single mode, or all modes from l = l_min to l = l_mode, with |m| m_mode. This is controlled by the mode_type parameter, which can be set to "all_modes" or "specific_mode". The parameters l_min and l_mode specify the lowest and highest modes to compute. The parameter m_mode specifies up to which value of m to compute. When using mode_type = "specific_mode", the mode l and m are given by l_mode and m_mode respectively.

4.3.2 Variable options

Several variable-specific options can be listed as tags in the variables parameter:

Multipole::variables = "<imp>::<var>{<tagname> = <tagvalue> ... } ..."

Valid tags are:

cmplx

A string giving the fully qualified variable name for the imaginary part of a complex field, assuming that <imp>::<var> is the real part.

name

A string giving an alias to use for the decomposed variable in the output filename, for use in the case of complex variable when otherwise the name of the real part would be used, which might be confusing.

sw

The integer spin-weight of the spherical harmonic decomposition to use.

Strings should be enclosed in single quotes within the list of tags.

4.3.3 Complex variables

In order to decompose a complex quantity, Multipole currently requires that the field is stored in two separate CCTK_REAL grid functions, one for the real and one for the imaginary part. Suppose the complex function u is stored in two gridfunctions u_re and u_im. In order to correctly decompose u, specify the real variable in the variables parameter, and use the tag cmplx to specify the name of its imaginary companion:

Multipole::variables = "MyThorn::u_re{cmplx = ’MyThorn::u_im’}"

4.3.4 Renaming variables

In some cases, you might want the name of the variable in the output filename to be different to the name of the grid function. This can be done by setting the name tag of the variable:

Multipole::variables = "MyThorn::u{name = ’myfunction’}"

For example, in the case of a complex variable where the output file contains the name of the real part, you can rename it as follows:

Multipole::variables = "MyThorn::u_re{cmplx = ’MyThorn::u_im’ name = ’u’}"

4.3.5 Spin weight

Depending on the nature of the field to be decomposed, a spin-weight other than 0 might be required in the spherical harmonic basis. Use the tag sw for this

Multipole::variables = "MyThorn::u_re{cmplx = ’MyThorn::u_im’ name = ’u’ sw = -2}"

4.3.6 Interpolator options

The interpolator to be used can be specified in the interpolator_name parameter, and a string containing interpolator parameters can be specified in the interpolator_pars parameter. See the interpolator (for example AEIThorns/AEILocalInterp) documentation for details of interpolators available and their options.

4.3.7 Output

When used with mesh-refinement, it is common to require mode decomposition less frequently than every iteration. The parameter out_every can be used to control this. 1D and 2D output of the coordinate spheres can be enabled using out_1d_every and out_2d_every.

4.4 Interaction With Other Thorns

This thorn obtains grid function data via the standard Cactus interpolator interface. To use this, one needs the parallel driver (for example PUGHInterp or CarpetInterp) as well as the low-level interpolator (e.g. AEILocalInterp).

4.5 Examples

To use this thorn with WeylScal4 to compute modes of the complex Ψ4 variable, one could use the following example:

ActiveThorns = ".... WeylScal4 CarpetInterp AEILocalInterp Multipole"  
 
Multipole::nradii    = 3  
Multipole::radius[0] = 10  
Multipole::radius[1] = 20  
Multipole::radius[2] = 30  
Multipole::variables = "WeylScal4::Psi4r{sw=-2 cmplx=’WeylScal4::Psi4i’}"  
Multipole::l_mode    = 4  
Multipole::m_mode    = 4

5 History

This thorn was developed in the Penn State Numerical Relativity group and contributed to the Einstein Toolkit.

5.1 Acknowledgements

This thorn was written by Ian Hinder and Andrew Knapp, with contributions from Eloisa Bentivegna and Shaun Wood.

References

[1]   J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan. Spin s spherical harmonics and edth. J. Math. Phys., 8:2155, 1967.

6 Parameters




coord_system
Scope: restricted  STRING



Description: What is the coord system?



Range   Default: cart3d
.*
Any smart string will do






enable_test
Scope: restricted  BOOLEAN



Description: whether to set a spherical harmonic in the ’harmonic’ grid functions



  Default: no






hdf5_chunk_size
Scope: restricted  INT



Description: How many iterations to preallocate in extensible HDF5 datasets



Range   Default: 200
1:*
Any integer






integration_method
Scope: restricted  KEYWORD



Description: How to do surface integrals



Range   Default: midpoint
midpoint
Midpoint rule (1st order)
Simpson
Simpson’s rule (4th order) [requires even ntheta and nphi]
trapezoidal
Trapezoidal rule (2nd order)
DriscollHealy
Driscoll & Healy (exponentially convergent) [requires even ntheta]






interpolator_name
Scope: restricted  STRING



Description: Which interpolator should I use



Range   Default: Hermite polynomial interpolation
.+
Any nonempty string






interpolator_pars
Scope: restricted  STRING



Description: Parameters for the interpolator



Range  Default: order=3 boundary_off_centering_tolerance={0.0 0.0 0.0 0.0 0.0 0.0} boundary_extrapolation_tolerance={0.0 0.0 0.0 0.0 0.0 0.0}
.*
”Any string that Util_TableSetFromStr ing() will take”






l_max
Scope: restricted  INT



Description: The maximum l mode to extract



Range   Default: 2
0:*
l >= 0






l_min
Scope: restricted  INT



Description: all modes: above which l mode to calculate/ specific mode: which l mode to extract



Range   Default: -1
-1:*
Deprecated






l_mode
Scope: restricted  INT



Description: The maximum l mode to extract



Range   Default: -1
-1:*
Deprecated






m_mode
Scope: restricted  INT



Description: all modes: Up to which m mode to calculate/ specific mode: which m mode to extract



Range   Default: -100
-100:*
Deprecated






mode_type
Scope: restricted  KEYWORD



Description: Which type of mode extraction do we have



Range   Default: deprecated
all modes
Extract all modes up to (l_mode, m_mode).
specific mode
Select one specific (l_mode, m_mode) mode
deprecated
Deprecated






nphi
Scope: restricted  INT



Description: How many points in the phi direction?



Range   Default: 100
1:*






nradii
Scope: restricted  INT



Description: How many extraction radii?



Range   Default: 1
0:100






ntheta
Scope: restricted  INT



Description: How many points in the theta direction?



Range   Default: 50
0:*
Positive please






out_1d_every
Scope: restricted  INT



Description: How often to output 1d data



Range   Default: (none)
no output
1:*
output every to many iterations






out_dir
Scope: restricted  STRING



Description: Output directory for Extract’s files, overrides IO::out_dir



Range   Default: (none)
.+
A valid directory name
An empty string to choose the default from IO::out_dir






out_every
Scope: restricted  INT



Description: How often to output



Range   Default: 1
no output
1:*
output every to many iterations






output_ascii
Scope: restricted  BOOLEAN



Description: Output a simple ASCII file for each mode at each radius



  Default: yes






output_hdf5
Scope: restricted  BOOLEAN



Description: Output an HDF5 file for each variable containing one dataset per mode at each radius



  Default: no






radius
Scope: restricted  REAL



Description: The radii for extraction



Range   Default: 0.0
0.0:*
Please keep it in the grid






test_l
Scope: restricted  INT



Description: which mode to put into the test variables



Range   Default: 2
*
Any integer






test_m
Scope: restricted  INT



Description: which mode to put into the test variables



Range   Default: 2
*
Any integer






test_mode_proportional_to_r
Scope: restricted  BOOLEAN



Description: whether the test spherical harmonic coefficient is proportional to the radial coordinate



  Default: no






test_sw
Scope: restricted  INT



Description: which spin weight to put into the test variables



Range   Default: -2
*
Any integer






variables
Scope: restricted  STRING



Description: What variables to decompose



Range   Default: (none)
.*
A list of variables






verbose
Scope: restricted  BOOLEAN



Description: Output detailed information about what is happening



  Default: no






io_out_dir
Scope: shared from IO STRING



7 Interfaces

General

Implements:

multipole

Inherits:

grid

Grid Variables

7.0.1 PUBLIC GROUPS




  Group Names     Variable Names     Details   




harmonics   compact0
harmonic_re   descriptionSpherical harmonics
harmonic_im   dimensions3
  distributionDEFAULT
  group typeGF
  timelevels1
 variable typeREAL




test_integration_convergence_orders   compact0
test_midpoint_convergence_order   descriptionTest integration convergence orders
test_trapezoidal_convergence_order  dimensions0
test_simpson_convergence_order   distributionCONSTANT
  group typeSCALAR
  timelevels1
 variable typeREAL




test_integration_results   compact0
test_midpoint_result_low   descriptionTest integration results
test_midpoint_result_high   dimensions0
test_trapezoidal_result_low   distributionCONSTANT
test_trapezoidal_result_high   group typeSCALAR
test_simpson_result_low   timelevels1
test_simpson_result_high  variable typeREAL




test_integration_symmetries   compact0
test_midpoint_pi_symmetry   descriptionTest integration symmetries
test_trapezoidal_pi_symmetry   dimensions0
test_simpson_pi_symmetry   distributionCONSTANT
test_driscollhealy_pi_symmetry   group typeSCALAR
  timelevels1
 variable typeREAL




8 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinAnalysis/Multipole. 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:
  harmonics[1]
  test_integration_convergence_orders
  test_integration_results
  test_integration_symmetries
   

Scheduled Functions

CCTK_ANALYSIS (conditional)

  multipole_calc

  calculate multipoles

 

 After: calc_np
   psikadelia
   accelerator_copyback
 Language:c
 Options: global
 Type: function

CCTK_INITIAL (conditional)

  multipole_setharmonic

  populate grid functions with spherical harmonics

 

 Language:c
 Type: function

CCTK_PARAMCHECK (conditional)

  multipole_paramcheck

  check multipole parameters

 

 Language:c
 Options: global
 Type: function

CCTK_PARAMCHECK (conditional)

  multipole_testintegrationconvergence

  test convergence of integration

 

 Language:c
 Type: function

CCTK_PARAMCHECK (conditional)

  multipole_testintegrationsymmetry

  test symmetry of integration

 

 Language:c
 Type: function