LeanBSSNMoL: An Einstein Toolkit thorn for evolving Einstein’s equations in the BSSN formalism with the MoL thorn

Miguel Zilhão and Helvi Witek

June 30, 2023

Abstract

LeanBSSNMoL solves Einstein’s equations in the BSSN formalism with up to 6th order accurate finite-difference stencils. It relies on the MoL thorn for the time integration.

1 LeanBSSNMoL

The Lean code was first introduced in [1], to evolve vacuum spacetimes with the BSSN formalism. It has since been modified to run with the MoL thorn for the time integration and generalized to include matter terms through the TmunuBase thorn. It was made publicly available through the Canuda numerical relativity library [2], and has since been distributed also as a part of the Einstein Toolkit.

The bulk of the code is written in Fortran 90 and should be simple to follow – emphasis has been given to readability. The main part of the code, where the right-hand side of the evolution equations is computed, can be found in the file calc_bssn_rhs.F90. For the conformal factor the code uses the “W” version, i.e. \(W=\gamma ^{-1/6}\), and it employs the usual “1+log” and “Gamma-driver” gauge conditions.

2 Obtaining this thorn

LeanBSSNMoL is included with the Einstein Toolkit and can also be obtained through the Canuda numerical relativity library [2].

References

[1] U. Sperhake, “Binary black-hole evolutions of excision and puncture data,” Phys. Rev. D 76 (2007), 104015 doi:10.1103/PhysRevD.76.104015 [arXiv:gr-qc/0606079 [gr-qc]].

[2] H. Witek, M. Zilhao, G. Bozzola, C.-H. Cheng, A. Dima, M. Elley, G. Ficarra, T. Ikeda, R. Luna, C. Richards, N. Sanchis-Gual, H. Okada da Silva. “Canuda: a public numerical relativity library to probe fundamental physics,” Zenodo (2023) doi: 10.5281/zenodo.3565474

3 Parameters


beta_alp	Scope: private	REAL

Description: Exponent for lapse in front of the Gammaî term in the shift

Range		Default: 0.0
:	Anything possible, default is zero


beta_f_delta1	Scope: private	REAL

Description: delta1 coefficient for the shift condition of 1702.01755

Range		Default: 0.0125
0:*	non-negative


beta_f_delta2	Scope: private	REAL

Description: delta2 coefficient for the shift condition of 1702.01755

Range		Default: 0.0005
0:*	non-negative


beta_gamma	Scope: private	REAL

Description: Coefficient in front of the Gammaî term in the shift

Range		Default: 0.75
:	Anything possible


calculate_constraints	Scope: private	BOOLEAN

Description: Calculate the BSSN constraints?

		Default: no


calculate_constraints_every	Scope: private	INT

Description: Calculate the BSSN constraints every N iterations

Range		Default: 1
:	0 or a negative value means never compute them


chi_gamma	Scope: private	REAL

Description: adding Yo-term to the gamma equation

Range		Default: 0.0
:	2/3 is a good value; the sign must be the same as betak,k


compute_rhs_at_initial	Scope: private	BOOLEAN

Description: Compute RHS after the initial data?

		Default: no


conf_fac_floor	Scope: private	REAL

Description: Minimal value of conformal factor

Range		Default: 1.0d-04
:	Any value possible


derivs_order	Scope: private	INT

Description: Order for derivatives

Range		Default: 4
4	4th order stencils
6	6th order stencils


eps_r	Scope: private	REAL

Description: Minimal value of radius for eta_transition

Range		Default: 1.0d-06
0:*	Any value possible


eta_beta	Scope: private	REAL

Description: Damping parameter in live shift

Range		Default: 1
0:*	non-negative


eta_transition	Scope: private	BOOLEAN

Description: Use an r-dependent eta?

		Default: no


eta_transition_r	Scope: private	REAL

Description: Damping parameter in live shift

Range		Default: 1
0:*	non-negative


impose_conf_fac_floor_at_initial	Scope: private	BOOLEAN

Description: Use floor value on initial data?

		Default: no


make_aa_tracefree	Scope: private	BOOLEAN

Description: Remove trace of aij after each timestep?

		Default: yes


n_aij	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for A_ij

Range		Default: 2
0:2	2 is reasonable


n_alpha	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for alpha

Range		Default: 1
0:2	1 is my guess


n_beta	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for beta

Range		Default: 1
0:2	1 is my guess


n_conf_fac	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for conf_fac

Range		Default: 1
0:2	1 is reasonable


n_gammat	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for Gammaî

Range		Default: 1
0:2	Maybe 1?


n_hij	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for h_ij

Range		Default: 1
0:2	1 is reasonable


n_trk	Scope: private	INT

Description: n power of outgoing boundary r fall off rate for A_ij

Range		Default: 2
0:2	2 is reasonable


precollapsed_lapse	Scope: private	BOOLEAN

Description: Initialize lapse as alp*psi-2}?

		Default: no


rescale_shift_initial	Scope: private	BOOLEAN

Description: Initialize shift as psi-2}beta ?

		Default: no


reset_dethh	Scope: private	BOOLEAN

Description: Reset determinant of conformal metric?

		Default: no


use_advection_stencils	Scope: private	BOOLEAN

Description: Use lop-sided stencils for advection derivs

		Default: yes


z_is_radial	Scope: private	BOOLEAN

Description: use with multipatch

		Default: no


zeta_alpha	Scope: private	REAL

Description: Coefficient in front of the ad1_alpha term in slicing

Range		Default: 1
:	Anything possible


zeta_beta	Scope: private	REAL

Description: Factor in front of ad1_beta in the live shift

Range		Default: 1
0:*	non-negative


leanbssn_maxnumconstrainedvars	Scope: restricted	INT

Description: The maximum number of constrained variables used by LeanBSSNMoL

Range		Default: 16
16:16	metric(6), extrinsic curvature(6), dtlapse(1) and dtshift(3)


leanbssn_maxnumevolvedvars	Scope: restricted	INT

Description: The maximum number of evolved variables used by LeanBSSNMoL

Range		Default: 21
21:21	lapse (1), shift(3), hmetric (6), hcurv(6), trK (1), conf_fac(1), Gamma (3)


leanbssn_maxnumsandrvars	Scope: restricted	INT

Description: The maximum number of save and restore variables used by LeanBSSNMoL

Range		Default: (none)
0:0	none


mol_num_constrained_vars	Scope: shared from METHODOFLINES	INT


mol_num_evolved_vars	Scope: shared from METHODOFLINES	INT


mol_num_saveandrestore_vars	Scope: shared from METHODOFLINES	INT

4 Interfaces

General

Implements:

leanbssnmol

Inherits:

admbase

tmunubase

boundary

coordgauge

Grid Variables

4.0.1 PUBLIC GROUPS


Group Names	Variable Names	Details

conf_fac	conf_fac	compact	0
		description	conformal factor
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” tensorweight=-0.33333333333333333333
		timelevels	3
		variable type	REAL

rhs_conf_fac	rhs_conf_fac	compact	0
		description	conformal factor
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” tensorweight=-0.33333333333333333333 prolongation=”none”
		timelevels	1
		variable type	REAL

hmetric		compact	0
	hxx	description	∖tilde gamma_ij
	hxy	dimensions	3
	hxz	distribution	DEFAULT
	hyy	group type	GF
	hyz	tags	tensortypealias=”DD_sym” tensorweight=-0.66666666666666666667
	hzz	timelevels	3
		variable type	REAL

rhs_hmetric		compact	0
	rhs_hxx	description	∖tilde gamma_ij
	rhs_hxy	dimensions	3
	rhs_hxz	distribution	DEFAULT
	rhs_hyy	group type	GF
	rhs_hyz	tags	tensortypealias=”DD_sym” tensorweight=-0.66666666666666666667 prolongation=”none”
	rhs_hzz	timelevels	1
		variable type	REAL

hcurv		compact	0
	axx	description	∖tilde a_ij
	axy	dimensions	3
	axz	distribution	DEFAULT
	ayy	group type	GF
	ayz	tags	tensortypealias=”DD_sym” tensorweight=-0.66666666666666666667
	azz	timelevels	3
		variable type	REAL

rhs_hcurv		compact	0
	rhs_axx	description	∖tilde a_ij
	rhs_axy	dimensions	3
	rhs_axz	distribution	DEFAULT
	rhs_ayy	group type	GF
	rhs_ayz	tags	tensortypealias=”DD_sym” tensorweight=-0.66666666666666666667 prolongation=”none”
	rhs_azz	timelevels	1
		variable type	REAL


Group Names	Variable Names	Details

trk		compact	0
	tracek	description	Tr(K)
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” tensorweight=0
		timelevels	3
		variable type	REAL

rhs_trk		compact	0
	rhs_tracek	description	Tr(K)
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” tensorweight=0 prolongation=”none”
		timelevels	1
		variable type	REAL

gammat		compact	0
	gammatx	description	∖tilde ∖Gammaî
	gammaty	dimensions	3
	gammatz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”U” tensorweight=0.66666666666666666667 tensorspecial=”Gamma”
		timelevels	3
		variable type	REAL

rhs_gammat		compact	0
	rhs_gammatx	description	∖tilde ∖Gammaî
	rhs_gammaty	dimensions	3
	rhs_gammatz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”U” tensorweight=0.66666666666666666667 tensorspecial=”Gamma” prolongation=”none”
		timelevels	3
		variable type	REAL

rhs_lapse		compact	0
	rhs_alp	description	lapse function
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” prolongation=”none”
		timelevels	1
		variable type	REAL

rhs_shift		compact	0
	rhs_betax	description	shift vector
	rhs_betay	dimensions	3
	rhs_betaz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”U” prolongation=”none”
		timelevels	1
		variable type	REAL


Group Names	Variable Names	Details

ham		compact	0
	hc	description	Hamiltonian constraint
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” tensorweight=0
		timelevels	3
		variable type	REAL

mom		compact	0
	mcx	description	momentum constraints
	mcy	dimensions	3
	mcz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”D” tensorweight=0
		timelevels	3
		variable type	REAL

Uses header:

Slicing.h

5 Schedule

This section lists all the variables which are assigned storage by thorn Lean/LeanBSSNMoL. 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:
	conf_fac[3] rhs_conf_fac[1]
	hmetric[3] rhs_hmetric[1]
	hcurv[3] rhs_hcurv[1]
	trk[3] rhs_trk[1]
	gammat[3] rhs_gammat[1]
	rhs_lapse[1]
	rhs_shift[1]
	ham[3]
	mom[3]

Scheduled Functions

CCTK_PARAMCHECK (conditional)

lean_paramcheck

check lean parameters for consistency

	Language:	c
	Type:	function

CCTK_STARTUP (conditional)

leanbssn_registerslicing

	Language:	c
	Options:	meta
	Type:	function

CCTK_POSTINITIAL (conditional)

leanbssn_calc_bssn_rhs_bdry

mol boundary rhs calculation

	After:	leanbssn_calcrhs
	Language:	fortran
	Type:	function

CCTK_POSTINITIAL (conditional)

leanbssn_calc_bssn_rhs_bdry_sph

mol boundary rhs calculation in spherical coordinates

	Language:	fortran
	Type:	function

MoL_CalcRHS (conditional)

leanbssn_calc_bssn_rhs_bdry

mol boundary rhs calculation

	After:	leanbssn_calcrhs
	Language:	fortran
	Type:	function

MoL_RHSBoundaries (conditional)

leanbssn_calc_bssn_rhs_bdry_sph

mol boundary rhs calculation in spherical coordinates

	Language:	fortran
	Type:	function

MoL_PostStep (conditional)

leanbssn_reset_detmetric

reset dethh = 1

	Before:	leanbssn_boundaries
	Language:	fortran
	Type:	function

MoL_PostStep (conditional)

leanbssn_remove_tra

remove trace of a

	After:	reset_detmetric
	Before:	leanbssn_boundaries
	Language:	fortran
	Type:	function

MoL_PostStep (conditional)

leanbssn_impose_conf_fac_floor

make sure conformal factor does not drop below specified value

	Before:	leanbssn_boundaries
	Language:	fortran
	Type:	function

MoL_PostStep (conditional)

leanbssn_boundaries

mol boundary enforcement

	Language:	fortran
	Options:	level
	Sync:	admbase::lapse
		admbase::shift
		leanbssnmol::conf_fac
		leanbssnmol::hmetric
		leanbssnmol::hcurv
		leanbssnmol::trk
		leanbssnmol::gammat
	Type:	function

MoL_PostStep (conditional)

applybcs

apply boundary conditions

	After:	leanbssn_boundaries
	Type:	group

MoL_PostStep (conditional)

leanbssn_bssn2adm

convert variables back to the adm ones

	After:	leanbssn_applybcs
	Before:	admbase_setadmvars
	Language:	fortran
	Options:	local
	Type:	function

CCTK_BASEGRID (conditional)

leanbssn_symmetries

	Language:	fortran
	Options:	global
	Type:	function

CCTK_ANALYSIS (conditional)

leanbssn_constraints

compute constraints

	Language:	fortran
	Type:	group

LeanBSSN_constraints (conditional)

leanbssn_bssn_constraints

compute constraints

	Language:	fortran
	Type:	function

LeanBSSN_constraints (conditional)

leanbssn_constraints_boundaries

enforce symmetry bcs in constraint computation

	After:	leanbssn_bssn_constraints
	Language:	fortran
	Options:	level
	Sync:	leanbssnmol::ham
		leanbssnmol::mom
	Type:	function

LeanBSSN_constraints (conditional)

applybcs

apply boundary conditions

	After:	leanbssn_constraints_boundaries
	Type:	group

CCTK_BASEGRID (conditional)

leanbssn_zero_rhs

set all rhs functions to zero to prevent spurious nans

	After:	leanbssn_symmetries
	Language:	fortran
	Type:	function

CCTK_INITIAL (conditional)

leanbssn_adm2bssn

convert initial data into bssn variables

	After:	admbase_postinitial
	Language:	fortran
	Options:	local
	Type:	function

CCTK_INITIAL (conditional)

leanbssn_boundaries

boundary enforcement

	After:	leanbssn_adm2bssn
	Language:	fortran
	Options:	level
	Sync:	leanbssnmol::gammat
	Type:	function

CCTK_INITIAL (conditional)

applybcs

apply boundary conditions

	After:	leanbssn_boundaries
	Type:	group

MoL_Register (conditional)

leanbssn_registervars

	Language:	c
	Options:	meta
	Type:	function

MoL_CalcRHS (conditional)

leanbssn_calc_bssn_rhs

mol rhs calculation

	Language:	fortran
	Type:	function

CCTK_POSTINITIAL (conditional)

leanbssn_calc_bssn_rhs

mol rhs calculation

	Before:	leanbssn_boundaries
	Language:	fortran
	Type:	function

Aliased Functions

Alias Name:	Function Name:
ApplyBCs	LeanBSSN_ApplyBCs
LeanBSSN_calc_bssn_rhs	LeanBSSN_CalcRHS
LeanBSSN_calc_bssn_rhs_bdry	LeanBSSN_CalcRHS_Bdry
LeanBSSN_calc_bssn_rhs_bdry_sph	LeanBSSN_CalcRHS_Bdry_Sph