StaticConformal

Tom Goodale et al

Date

Abstract

Base thorn to provide the variables for the static conformal factor

1 Purpose

This thorn provides the variables defining a static conformal factor which is used to transform the physical metric. If this thorn is active and the ADMBase::metric_type parameter is set to static conformal, then the ADMBase::g... variables are the conformal values as opposed to the physical values.

The transformation is

gijphysical = ψ4g ijconformal

The extrinsic curvature is not transformed.

Memory is provided for the conformal factor psi, its first derivatives psix, psiy, psiz, and its second derivatives psixx, psixy, psixz, psiyy, psiyz, and psizz depending on the setting of the conformal_storage parameter.

Note that the first and second “derivative” grid functions have an additional factor of 1ψ normalisation since this is the most common use of the derivative. I.e., the grid functions are

psi = ψ, psix = ψxψ,etc psixx = ψijψetc

Thorns need to check the value of the grid scalar conformal_state to determine how many levels of these variables have actually been calculated before using the conformal factor:

conformal_state=0
No conformal factor has been calculated — thorns may assume the conformal factor is 1 at all points. (I.e., the metric is physical.)
conformal_state=1
The conformal factor has been calulated, but no derivatives.
conformal_state=2
The conformal factor and its first derivatives have been calculated.
conformal_state=3
The conformal factor and its first and second derivatives have been calculated.

Note that this means that if you only want to know whether psi contains the values for the conformal factor you can check for conformal_state > 0.

2 Utilities

StaticConformal provides aliased functions to convert between physical and conformal 3-metric values. It is very important to understand that these functions apply the conversion in place. That is, if gxx contains the conformal metric value, when the routine is exited it will now contain the physical metric value. These functions do not change the value of conformal_state and should be used with due care. (These functions are for example used by some analysis thorns who work only with the physical metric, they apply the transformation on entry to the analysis routine and switch it back on exit).

Convert from conformal to physical:
 
subroutine ConfToPhysInPlace (nx, ny, nz,  
                              psi,  
                              gxx, gxy, gxz, gyy, gyz, gzz)  
   implicit none  
   CCTK_INT,                         intent(in)    :: nx, ny, nz  
   CCTK_REAL, dimension(nx, ny, nz), intent(in)    :: psi  
   CCTK_REAL, dimension(nx, ny, nz), intent(inout) :: gxx, gxy, gxz, gyy, gyz, gzz  
end subroutine ConfToPhysInPlace

Convert from physical to conformal:
 
subroutine PhysToConfInPlace (nx, ny, nz,  
                              psi,  
                              gxx, gxy, gxz, gyy, gyz, gzz)  
   implicit none  
   CCTK_INT,                         intent(in)    :: nx, ny, nz  
   CCTK_REAL, dimension(nx, ny, nz), intent(in)    :: psi  
   CCTK_REAL, dimension(nx, ny, nz), intent(inout) :: gxx, gxy, gxz, gyy, gyz, gzz  
end subroutine ConfToPhysInPlace

3 Comments

The StaticConformal thorn itself does not calculate any conformal factor, but does initialise the conformal_state variable to 0.

4 Parameters




conformal_storage
Scope: restricted  KEYWORD



Description: How much conformal storage do we have ?



Range   Default: factor+derivs+2nd derivs
factor
Just the conformal factor
factor+derivs
Conformal factor plus first derivatives
see [1] below
Conformal factor plus first and second derivatives



[1]

factor+derivs+2nd derivs




metric_type
Scope: shared from ADMBASE  KEYWORD



Extends ranges:



static conformal
Metric is conformal with static conformal factor, extrinsic curvature is physical



5 Interfaces

General

Implements:

staticconformal

Inherits:

grid

Grid Variables

5.0.1 PUBLIC GROUPS




  Group Names    Variable Names    Details   




conformal_state   compact0
conformal_state   dimensions0
  distributionCONSTANT
  group typeSCALAR
  timelevels1
 variable typeINT




confac   compact0
psi   descriptionConformal factor
  dimensions3
  distributionDEFAULT
  group typeGF
  tagstensortypealias=”Scalar” Prolongation=”None” InterpNumTimelevels=1
  timelevels1
 variable typeREAL




confac_1derivs   compact0
psix   descriptionFirst spatial derivatives of conformal factor divided by psi
psiy   dimensions3
psiz   distributionDEFAULT
  group typeGF
  tagstensortypealias=”U” Prolongation=”None” InterpNumTimelevels=1
  timelevels1
 variable typeREAL




confac_2derivs   compact0
psixx   descriptionSecond spatial derivatives of conformal factor divided by psi
psixy   dimensions3
psixz   distributionDEFAULT
psiyy   group typeGF
psiyz   tagstensortypealias=”dd_sym” Prolongation=”None” InterpNumTimelevels=1
psizz   timelevels1
 variable typeREAL




Provides:

ConfToPhysInPlace to

PhysToConfInPlace to

6 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinBase/StaticConformal. 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

 

Always: Conditional:
conformal_stateconfac[1]
  confac[1]
  confac_1derivs[1]
  confac[1]
  confac_1derivs[1]
  confac_2derivs[1]
   

Scheduled Functions

CCTK_INITIAL

  staticconformal_initialisestate

  set the conformal_state variable to 0

 

 Before: admbase_initialdata
  Language:c
 Type: function