Baikal: An Einstein Toolkit thorn for solving Einstein’s equations in the BSSN formalism, in Cartesian coordinates

Zachariah B. Etienne <zachetie *at* gmail *dot* com>

May 18, 2020

Abstract

Baikal solves Einstein’s equations of general relativity in the BSSN formalism. Given that the complications and super-short-lengthscale variations usually associated with the stress-energy tensor $T^{μ ν}$ in strongly curved spacetimes, this thorn implements finite-difference derivatives up to and including 4th order, but no higher. All codes within this thorn were generated using the Python-based NRPy+ infrastructure, which may be found at http://nrpyplus.net/(alt link: https://github.com/zachetienne/nrpytutorial).

Both Baikal and its sister thorn BaikalVacuum are fully documented in Jupyter notebooks within NRPy+, and the latest documentation for BaikalETK (the name of the combined Baikal / BaikalVacuum project) may be found in the Tutorial-BaikalETK.ipynb Jupyter notebook here:

https://github.com/zachetienne/nrpytutorial

Also the NRPy+ Jupyter documentation can be navigated via the following link:

https://nbviewer.jupyter.org/github/zachetienne/nrpytutorial/blob/master/Tutorial-BaikalETK.ipynb

1 Planned features

Currently Baikal / BaikalVacuum do not officially support numerical grids containing symmetry assumptions, but appropriate symmetries should be registered. Users are encouraged to proceed with caution when using symmetries. Future work will include adding official support for symmetries.
Baikal / BaikalVacuum do not yet support an extensive variety of gauge conditions, instead defaulting to the tried-and-true standard moving puncture gauge conditions ( $1 + \log$ lapse combined with the “Gamma”-driving shift [though note that Baikal / BaikalVacuum refer to the “Gamma” variable as the “Lambda” variable]). Additional gauge condition options will eventually be added.
Baikal / BaikalVacuum do not support Llama-based multiblock (e.g., cubed sphere) grids, and adding this support is currently not a high priority.

2 Parameters


diss_strength	Scope: restricted	REAL

Description: diss_strength (see NRPy+ for parameter definition)

Range		Default: 0.1
:	All values accepted. NRPy+ does not restrict the allowed ranges of parameters yet.


eta	Scope: restricted	REAL

Description: eta (see NRPy+ for parameter definition)

Range		Default: 2.0
:	All values accepted. NRPy+ does not restrict the allowed ranges of parameters yet.


fd_order	Scope: restricted	INT

Description: Finite-differencing order

Range		Default: 4
2:2	”finite-differencing order = 2”
4:4	”finite-differencing order = 4”


lapse_floor	Scope: restricted	REAL

Description: lapse_floor (see NRPy+ for parameter definition)

Range		Default: 1e-15
:	All values accepted. NRPy+ does not restrict the allowed ranges of parameters yet.


pi	Scope: restricted	REAL

Description: PI (see NRPy+ for parameter definition)

Range		Default: 3.14159265358979323846264338327950288
:	All values accepted. NRPy+ does not restrict the allowed ranges of parameters yet.


dtlapse_evolution_method	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

Baikal


dtshift_evolution_method	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

Baikal


evolution_method	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

Baikal


lapse_evolution_method	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

Baikal


shift_evolution_method	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

Baikal

3 Interfaces

General

Implements:

baikal

Inherits:

admbase

boundary

grid

methodoflines

tmunubase

Grid Variables

3.0.1 PUBLIC GROUPS


Group Names	Variable Names	Details

evol_variables		compact	0
	aDD00GF	description	BSSN evolved gridfunctions
	aDD01GF	dimensions	3
	aDD02GF	distribution	DEFAULT
	aDD11GF	group type	GF
	aDD12GF	timelevels	3
	aDD22GF	variable type	REAL

evol_variables_rhs		compact	0
	aDD00_rhsGF	description	right-hand-side storage for BSSN evolved gridfunctions
	aDD01_rhsGF	dimensions	3
	aDD02_rhsGF	distribution	DEFAULT
	aDD11_rhsGF	group type	GF
	aDD12_rhsGF	tags	InterpNumTimelevels=1 prolongation=”none”
	aDD22_rhsGF	timelevels	1
	alpha_rhsGF	variable type	REAL

aux_variables		compact	0
	HGF	description	Auxiliary gridfunctions for BSSN diagnostics
	MU0GF	dimensions	3
	MU1GF	distribution	DEFAULT
	MU2GF	group type	GF
		timelevels	3
		variable type	REAL

auxevol_variables		compact	0
	RbarDD00GF	description	Auxiliary gridfunctions needed for evaluating the BSSN RHSs
	RbarDD01GF	dimensions	3
	RbarDD02GF	distribution	DEFAULT
	RbarDD11GF	group type	GF
	RbarDD12GF	tags	InterpNumTimelevels=1 prolongation=”none”
	RbarDD22GF	timelevels	1
	T4UU00GF	variable type	REAL

Uses header:

Symmetry.h

Boundary.h

4 Schedule

This section lists all the variables which are assigned storage by thorn WVUThorns/Baikal. 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:
ADMBase::metric[1] ADMBase::curv[1] ADMBase::lapse[1] ADMBase::shift[1]
evol_variables[3]
evol_variables_rhs[1]
aux_variables[3]
auxevol_variables[1]

Scheduled Functions

CCTK_STARTUP

baikal_banner

output ascii art banner

	Language:	c
	Options:	meta
	Type:	function

CCTK_STARTUP

baikal_registerslicing

	After:	baikal_banner
	Language:	c
	Options:	meta
	Type:	function

MoL_CalcRHS

baikal_newrad

newrad boundary conditions, scheduled right after rhs eval.

	After:	baikal_rhs
	Language:	c
	Type:	function

MoL_PostStep

baikal_floor_the_lapse

set lapse = max(lapse_floor, lapse)

	Before:	baikal_enforce_detgammabar_constraint
		bc_update
	Language:	c
	Type:	function

MoL_PostStep

baikal_enforce_detgammabar_constraint

enforce detgammabar = detgammahat (= 1 in cartesian)

	Before:	bc_update
	Language:	c
	Type:	function

MoL_PostStep

baikal_boundaryconditions_evolved_gfs

apply boundary conditions and perform amr+interprocessor synchronization

	Language:	c
	Options:	level
	Sync:	evol_variables
	Type:	function

MoL_PostStep

applybcs

group for applying boundary conditions

	After:	baikal_boundaryconditions_evolved_gfs
	Type:	group

MoL_PostStep

baikal_bssn_to_adm

perform bssn-to-adm conversion. useful for diagnostics.

	After:	baikal_applybcs
	Before:	admbase_setadmvars
	Language:	c
	Options:	local
	Type:	function

MoL_PseudoEvolution

baikal_driver_bssn_t4uu

mol_pseudoevolution: compute t4uu, needed for bssn constraints

	Before:	baikal_bssn_constraints
	Language:	c
	Options:	local
	Type:	function

MoL_PseudoEvolution

baikal_bssn_constraints

compute bssn (hamiltonian and momentum) constraints

	Language:	c
	Options:	local
	Type:	function

MoL_PseudoEvolution

baikal_boundaryconditions_aux_gfs

enforce symmetry bcs in constraint computation

	After:	baikal_bssn_constraints
	Language:	c
	Options:	level
	Sync:	aux_variables
	Type:	function

MoL_PseudoEvolution

baikal_bssn_to_adm

perform bssn-to-adm conversion in mol_pseudoevolution. needed for proper hydrobase integration.

	After:	baikal_boundaryconditions_aux_gfs
	Language:	c
	Options:	local
	Type:	function

CCTK_BASEGRID

baikal_symmetry_registration

	Language:	c
	Options:	global
	Type:	function

MoL_PseudoEvolution

applybcs

apply boundary conditions

	After:	baikal_boundaryconditions_aux_gfs
	Type:	group

CCTK_BASEGRID

baikal_zero_rhss

idea from lean: set all rhs functions to zero to prevent spurious nans

	After:	baikal_symmetry_registration
	Language:	c
	Type:	function

CCTK_INITIAL

baikal_adm_to_bssn

convert initial data into bssn variables

	After:	admbase_postinitial
	Language:	c
	Options:	local
	Sync:	evol_variables
	Type:	function

CCTK_INITIAL

applybcs

apply boundary conditions

	After:	baikal_adm_to_bssn
	Type:	group

MoL_Register

baikal_mol_registration

	Language:	c
	Options:	meta
	Type:	function

MoL_CalcRHS

baikal_driver_bssn_t4uu

mol: compute t4uu, needed for bssn rhss.

	Before:	baikal_bssn_to_adm
	Language:	c
	Type:	function

MoL_CalcRHS

baikal_driver_pt1_bssn_ricci

mol: compute ricci tensor

	Before:	baikal_rhs
	Language:	c
	Type:	function

MoL_CalcRHS

baikal_driver_pt2_bssn_rhss

mol: evaluate bssn rhss

	After:	baikal_ricci
	Language:	c
	Type:	function

Aliased Functions

Alias Name:	Function Name:
ApplyBCs	Baikal_ApplyBCs
Baikal_driver_BSSN_T4UU	Baikal_T4UU
Baikal_driver_pt1_BSSN_Ricci	Baikal_Ricci
Baikal_driver_pt2_BSSN_RHSs	Baikal_RHS