1 Introduction

TwoPuncturesX is the CarpetX port of TwoPunctures. It provides identical physics and parameters; the differences are entirely in the Cactus infrastructure layer:

• ADMBaseX instead of ADMBase: metric and extrinsic curvature are stored as CCTK_REAL grid functions under the ADMBaseX interface (gxx, kxx, alp, etc.).

• CoordinatesX for coordinates: the physical grid coordinates vcoordx, vcoordy, vcoordz are obtained from CoordinatesX rather than from the Carpet grid.

• Thread safety: CarpetX may call a scheduled function concurrently from multiple OpenMP threads. The one-time spectral solve is guarded by a named OpenMP critical section so that it is performed exactly once regardless of how many threads enter the function simultaneously.

• Physical metric storage: TwoPuncturesX always fills the physical (non-conformal) metric. The conformal factor and its derivatives that TwoPunctures optionally stores via StaticConformal are not available under CarpetX.

For full details of the underlying physics and algorithm, see the TwoPunctures documentation and [3].

2 Physical System

The physics is identical to TwoPunctures. The conformal decomposition

\begin {equation} \psi = 1 + \frac {m_+}{2 r_+} + \frac {m_-}{2 r_-} + u \end {equation}

is used, where \(m_\pm \) are the bare masses of the two punctures at \((\pm b, 0, 0)\), \(r_\pm \) are distances from the respective punctures, and \(u\) is a smooth correction solved spectrally. The Bowen-York extrinsic curvature [1] encodes linear momenta and spins.

The output metric is the physical metric \(\gamma _{ij} = \psi ^4 f_{ij}\) (conformally flat), not the conformal metric.

3 Numerical Implementation

3.1 Shared Solver

TwoPuncturesX reuses the spectral solver from TwoPunctures by including the header TP_Guts.h directly into the scheduling function body. This header contains the full solve-and-interpolate logic. Because TwoPunctures must be present in the same build, the compiler sees both thorns’ object files and can share the Newton solver and spectral machinery.

3.2 Thread Safety

Under CarpetX, the scheduled function TwoPuncturesX may be invoked from within an OpenMP parallel region. The expensive spectral solve must run exactly once. This is guaranteed by a static boolean flag did_setup and a named OpenMP critical section:

#pragma omp critical(TwoPunctures_setup)
  if (!did_setup) {
    // ... perform spectral solve ...
    did_setup = true;
  }

Using a named critical section avoids interference with the unnamed critical section inside Newton’s parallel reduction loop in Newton.cc.

3.3 Grid Fill

After the spectral solve the 3D grid is filled point-by-point using the same interpolation path as TwoPunctures (grid_setup_method = "Taylor expansion" or "evaluation"). The inner loop is parallelized with #pragma omp parallel for.

3.4 Scalar Grid Functions

CarpetX resets grid scalars at the start of each scheduled call. TwoPuncturesX therefore saves the computed scalars (mp, mm, mp_adm, mm_adm, E, J1, J2, J3) into static variables during setup and restores them on every call so that they remain available to other thorns.

4 Using This Thorn

4.1 Requirements

• EinsteinInitialData/TwoPunctures must be compiled into the same executable. TwoPuncturesX includes header files from that thorn and will fail to compile otherwise.

• The CarpetX infrastructure thorns (CarpetX, ADMBaseX, CoordinatesX) must be present.

4.2 Basic Usage

Add both EinsteinInitialData/TwoPunctures and EinsteinInitialData/TwoPuncturesX to your thorn list, then set:

ADMBaseX::initial_data  = "twopunctures"
ADMBaseX::initial_lapse = "twopunctures-antisymmetric"

TwoPuncturesX::par_b       = 3.0   # half separation (M)
TwoPuncturesX::par_m_plus  = 0.5
TwoPuncturesX::par_m_minus = 0.5
TwoPuncturesX::par_P_plus[1]  =  0.084
TwoPuncturesX::par_P_minus[1] = -0.084
TwoPuncturesX::npoints_A   = 30
TwoPuncturesX::npoints_B   = 30
TwoPuncturesX::npoints_phi = 16

The parameter names and semantics are the same as in TwoPunctures (see that thorn’s documentation for a full description of each parameter).

4.3 Lapse Options

The same lapse choices are available as in TwoPunctures: twopunctures-antisymmetric, twopunctures-averaged, psiˆn, W, and brownsville. See the TwoPunctures documentation for details. Note that the conformal-factor-based lapse options (psiˆn, brownsville) compute \(\psi \) locally during the grid fill; no separate conformal storage thorn is required.

4.4 ADM Mass Iteration

Setting give_bare_mass = no enables iterative determination of bare masses to match target ADM masses target_M_plus and target_M_minus, exactly as in TwoPunctures.

4.5 Interaction With Other Thorns

TwoPuncturesX writes to the ADMBaseX grid functions gxx, gyy, gzz, gxy, gxz, gyz, kxx, kyy, kzz, kxy, kxz, kyz, and alp. It also populates the scalar grid functions mp, mm, mp_adm, mm_adm, E, J1, J2, and J3.

Matter sources can be supplied via the same optional function interfaces Set_Rho_ADM and Set_Initial_Guess_for_u as in TwoPunctures; however, rescale_sources = yes with use_sources = yes is not yet supported and will trigger a run-time error.

5 History

TwoPuncturesX was derived from TwoPunctures to support the CarpetX AMR framework. The underlying spectral algorithm is unchanged from Ansorg, Brügmann & Tichy (2004) [3].

References

6 Parameters

initial_lapse_psi_exponent	Scope: restricted	REAL
Description: Exponent n for psin initial lapse profile
Range		Default: -2.0
(*:*)	Should be negative

7 Interfaces

General

Implements:

twopuncturesx

Inherits:

admbasex

coordinatesx

Grid Variables

7.0.1 PRIVATE GROUPS

7.0.2 PUBLIC GROUPS

Uses header:

TwoPunctures.h

TP_utilities.h

TP_params.h

TP_Guts.h

8 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/TwoPuncturesX. 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_PARAMCHECK (conditional)

  twopuncturesx_paramcheck

  check parameters and thorn needs

ADMBaseX_InitialData (conditional)

  twopuncturesx_group

  twopuncturesx initial data group

CCTK_INITIAL (conditional)

  twopuncturesx_group

  twopuncturesx initial data group

TwoPuncturesX_Group (conditional)

  twopuncturesx

  create puncture black hole initial data

TwoPuncturesX_Group (conditional)

  twopuncturesx_metadata

  output twopuncturesx metadata