A Multi-Patch Wave Toy

Erik Schnetter <schnetter@aei.mpg.de>
Luis Lehner <lehner@lsu.edu>
Manuel Tiglio <tiglio@lsu.edu>

December 12, 2024

1 Physical System

The massless wave equation for a scalar field \(\phi \) can be written as

∂μ(γμνdν) = 0

where \(\gamma ^{\mu \nu }=\sqrt {-g}g^{\mu \nu }\), and \(d_{\mu }\equiv \partial _{\mu }\phi \). Using the expressions \(\sqrt {-g} = \alpha \sqrt {h}\) and \begin {displaymath} g^{\mu \nu } = \left ( \begin {array}{cc} -1/\alpha ^2 & \beta ^i/\alpha ^2 \\ \beta ^j/\alpha ^2 & \gamma ^{ij} - \beta ^i \beta ^j/\alpha ^2 \end {array} \right ), \end {displaymath} where \(\gamma ^{ij}\) is the inverse of the three metric, the equation can be rewritten as

\begin {eqnarray} \dot {\phi } &=& \Pi , \\ \dot {\Pi } &=& \beta ^i\partial _i\Pi + \frac {\alpha }{\sqrt {h}}\partial _i\left (\frac {\sqrt {h}}{\alpha } \beta ^i\Pi + \frac {\sqrt {h}}{\alpha }H^{ij}d_j \right ) + \frac {\alpha }{\sqrt {h}}d_i\partial _t \left (\frac {\sqrt {h}\beta ^i}{\alpha } \right ) - \frac {\Pi \alpha }{\sqrt {h}}\partial _t \left ( \frac {\sqrt {h}}{\alpha } \right ), \\ \dot {d_i} &=& \partial _i \Pi \end {eqnarray}

with \(H^{ij} \equiv \alpha ^2 \gamma ^{ij} - \beta ^i \beta ^j\). The non-shift speed modes with respect to a boundary with normal \(n_i\) are

v± = λΠ + Hijn d i j

and the shift speed modes are \(d_A\), with \(A\) transversal directions.

The physical energy is

1∫ 1 √ -- E = 2 α∖left[Π2 + Hijdidj∖right] hdx3

and the way the equations above have been written this energy is not increasing in the stationary background case if homogeneous boundary conditions are given at outer boundaries, also at the discrete level (replacing above \(\partial _i\) by \(D_i\)).

The implementation of the field equations in the code differs slightly from the above. Assuming a stationary background and expanding out the derivative operator, we obtain the equations in the final form

\begin {eqnarray} \dot {\phi } &=& \Pi , \\ \dot {\Pi } &=& \beta ^i\partial _i\Pi + \frac {\alpha }{\sqrt {h}}\partial _i\left (\frac {\sqrt {h}}{\alpha } \beta ^i\Pi \right ) + \frac {\alpha }{\sqrt {h}}\partial _i\left (\frac {\sqrt {h}}{\alpha }H^{ij}\right ) \partial _j \phi + \frac {\alpha }{\sqrt {h}} \frac {\sqrt {h}}{\alpha } H^{ij} \partial _i\partial _j \phi . \end {eqnarray}

2 Non-linear addition to the multi-patch wave toy

Simple addition to the wave multi-patch toy to get started on this.

2.1 Physical System

This is just a non-linear wave equation obtained, a very minor modification to the wave equation thorn adding a different initial data and a slight modification to the right hand side. A reference to this is in a paper by Liebling to appear in Phys. Rev. D (2005). The non-linear wave equation is written as

∂ (γ μνd ) = ϕp u ν

where \(\gamma ^{\mu \nu }=\sqrt {-g}g^{\mu \nu }\), and \(d_{\mu }=\partial _{\mu }\phi \) and \(p\) must be an odd integer \(\ge 3\).

The initial data coded is given by

\begin {eqnarray} \phi &=& A e^{-(r_1 - R)^2/\delta ^2} \\ \Pi &=& \mu \phi _{,r} + \Omega ( y \phi _{,x} - x \phi _{,y} ) \\ d_i = \phi _{,i} \end {eqnarray}

with \(\tilde r^2 = \epsilon _x x^2 + \epsilon _y y^2 + z^2\).

The parameters used for this initial data are given some distinct names to avoid conflicts with existing ones and are as follows:

initial-data = GaussianNonLinear (choose the above mentioned initial data)
nonlinearrhs = turn on/off the right hand side for testing. a boolean variable.
powerrhs = power \(p\) above.
epsx = \(\epsilon _x\)
epsy = \(\epsilon _y\)
ANL = \(A\)
deltaNL = \(\delta \)
omeNL = \(\Omega \)
RNL = R

Note, as the solution is not known, one must set for now the incoming fields to \(0\). CPBC might one day be put... though who knows :-)

3 Formulations

\begin {eqnarray} U & = & \left [ \begin {array}{cccc} \rho & v_x & v_y & v_z \end {array} \right ]^T = \left [ \begin {array}{cc} \rho & v_i \end {array} \right ]^T \\ \partial _t U & = & A^i \partial _i U + \cdots \\ || n_i || & \ne & 1 \end {eqnarray}

3.1 \(dt\)

Setting \(\rho = \partial _t u\).

RHS:

\begin {eqnarray} H^{ij} & = & \alpha ^2 \gamma ^{ij} - \beta ^i \beta ^j \\ \partial _t u & = & \rho \\ \partial _t \rho & = & \beta ^i \partial _i \rho + \frac {\alpha }{\epsilon } \partial _i \frac {\epsilon }{\alpha } \left ( \beta ^i \rho + H^{ij} v_j \right ) \\ \partial _t v_i & = & \partial _i \rho \end {eqnarray}

Propagation matrix:

\begin {eqnarray} A^x & = & \left ( \begin {array}{cccc} 2 \beta ^x & - \beta ^x \beta ^x + \alpha ^2 \gamma ^{xx} & - \beta ^x \beta ^y + \gamma ^{xy} \alpha ^2 & -\beta ^x \beta ^z + \alpha ^2 \gamma ^{xz} \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {array} \right ) \end {eqnarray}

\begin {eqnarray} A^n & = & \left ( \begin {array}{cc} 2 \beta ^i n_i & \left ( - \beta ^i \beta ^j + \alpha ^2 \gamma ^{ij} \right ) n_i \\ n_i & 0 \end {array} \right ) \end {eqnarray}

Eigensystem:

\begin {eqnarray} \lambda _1 = 0 &, & w_1 = \left [ \begin {array}{cccc} 0 & 0 & 0 & 1 \end {array} \right ]^T \\ \lambda _2 = 0 &, & w_2 = \left [ \begin {array}{cccc} 0 & 0 & 1 & 0 \end {array} \right ]^T \\ \lambda _3 = \beta ^x - \alpha \sqrt {\gamma ^{xx}} &, & w_3 = \left [ \begin {array}{cccc} \beta ^x - \alpha \sqrt {\gamma ^{xx}} & \beta ^x \beta ^x + \alpha ^2 \gamma ^{xx} & \beta ^x \beta ^y + \alpha ^2 \gamma ^{xy} & \beta ^x \beta ^z + \alpha ^2 \gamma ^{xz} \end {array} \right ]^T \\ \lambda _4 = \beta ^x + \alpha \sqrt {\gamma ^{xx}} &, & w_4 = \left [ \begin {array}{cccc} \beta ^x + \alpha \sqrt {\gamma ^{xx}} & \beta ^x \beta ^x + \alpha ^2 \gamma ^{xx} & \beta ^x \beta ^y + \alpha ^2 \gamma ^{xy} & \beta ^x \beta ^z + \alpha ^2 \gamma ^{xz} \end {array} \right ]^T \end {eqnarray}

\begin {eqnarray} \lambda _t = 0 &, & w_t = \left [ \begin {array}{cc} 0 & t_i \end {array} \right ]^T \\ \lambda _\pm = \beta ^i n_i \pm \alpha \sqrt {\gamma ^{ij} n_i n_j} &, & w_\pm = \left [ \begin {array}{cccc} \beta ^i n_i \pm \alpha \sqrt {\gamma ^{ij} n_i n_j} & H^{ij} n_j \end {array} \right ]^T \end {eqnarray}

3.2 \(d0\)

Setting \(\rho = \mathcal {L}_n u\) with \(n_a = D_a t\), leading to \(\rho = \partial _0 u = (1/\alpha ) \partial _t u - (1/\alpha ) \beta ^i \partial _i u\).

RHS:

\begin {eqnarray} H^{ij} & = & \gamma ^{ij} \\ \partial _t u & = & \alpha \rho + \beta ^i v_i \\ \partial _t \rho & = & \beta ^i \partial _i \rho + \frac {1}{\epsilon } \partial _i \epsilon \alpha H^{ij} v_j + \frac {\rho }{\epsilon } \partial _i \epsilon \beta ^i \\ \partial _t v_i & = & \beta ^j \partial _j v_i + v_j \partial _i \beta ^j + \partial _i \alpha \rho \end {eqnarray}

Propagation matrix:

\begin {eqnarray} A^x & = & \left ( \begin {array}{cccc} \beta ^x & \alpha \gamma ^{xx} & \alpha \gamma ^{xy} & \alpha \gamma ^{xz} \\ \alpha & \beta ^x & 0 & 0 \\ 0 & 0 & \beta ^x & 0 \\ 0 & 0 & 0 & \beta ^x \end {array} \right ) \end {eqnarray}

\begin {eqnarray} A^n & = & \left ( \begin {array}{cc} \beta ^i n_i & \alpha \gamma ^{ij} n_i \\ \alpha n_i & \beta ^k n_k \delta _{ij} \end {array} \right ) \end {eqnarray}

Eigensystem:

\begin {eqnarray} \lambda _t = \beta ^i n_i &, & w_t = \left [ \begin {array}{cccc} 0 & - \gamma ^{ij} n_j n_k t^k + \gamma ^{jk} n_j n_k t^i \end {array} \right ]^T \\ \lambda _\pm = \beta ^i n_i \pm \alpha \sqrt { \gamma ^{ij} n_i n_j } &, & w_\pm = \left [ \begin {array}{cccc} \pm \sqrt { \gamma ^{ij} n_i n_j } & \gamma ^{ij} n_j \end {array} \right ]^T \end {eqnarray}

3.3 \(dk\)

(This section very probably contains errors, say Erik on 2005-04-13.)

Setting \(\rho = \mathcal {L}_k u\) with a “Killing” vector \(k^a = \partial _t x^a\), leading to \(\rho = (1/\alpha ) \partial _t u\).

RHS:

\begin {eqnarray} H^{ij} & = & \gamma ^{ij} - \frac { \beta ^i \beta ^j }{ \alpha ^2 } \\ \partial _t u & = & \alpha \rho \\ \partial _t \rho & = & \frac {\beta ^i}{\alpha } \partial _i \alpha \rho + \frac {1}{\epsilon } \partial _i \epsilon \left ( \beta ^i \rho + \alpha H^{ij} v_j \right ) \\ \partial _t v_i & = & \partial _i \alpha \rho \end {eqnarray}

Propagation matrix:

\begin {eqnarray} A^x & = & \left ( \begin {array}{cccc} 2 \beta ^x & \alpha \left ( - \frac {\beta ^x \beta ^x}{\alpha ^2} + \gamma ^{xx} \right ) & \alpha \left ( - \frac {\beta ^x \beta ^y}{\alpha ^2} + \gamma ^{xy} \right ) & \alpha \left ( - \frac {\beta ^x \beta ^z}{\alpha ^2} + \gamma ^{xz} \right ) \\ \alpha & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {array} \right ) \end {eqnarray}

Eigensystem:

\begin {eqnarray} \lambda _1 = 0 &, & w_1 = \left [ \begin {array}{cccc} 0 & - \beta ^x \beta ^z + \alpha ^2 \gamma ^{xz} & 0 & \beta ^x \beta ^x - \alpha ^2 \gamma ^{xx} \end {array} \right ]^T \\ \lambda _2 = 0 &, & w_2 = \left [ \begin {array}{cccc} 0 & - \beta ^x \beta ^y + \alpha ^2 \gamma ^{xy} & \beta ^x \beta ^x - \alpha ^2 \gamma ^{xx} & 0 \end {array} \right ]^T \\ \lambda _3 = \beta ^x - \alpha \sqrt {\gamma ^{xx}} &, & w_3 = \left [ \begin {array}{cccc} \beta ^x - \sqrt {\gamma ^{xx}} & \alpha & 0 & 0 \end {array} \right ]^T \\ \lambda _4 = \beta ^x + \alpha \sqrt {\gamma ^{xx}} &, & w_4 = \left [ \begin {array}{cccc} \beta ^x + \sqrt {\gamma ^{xx}} & \alpha & 0 & 0 \end {array} \right ]^T \end {eqnarray}

References

[1] Gioel Calabrese, Luis Lehner, Dave Neilsen, Jorge Pullin, Oscar Reula, Olivier Sarbach, Manuel Tiglio, Novel finite-differencing techniques for numerical relativity: application to black-hole excision, Class. Quantum Grav. 20, L245 (2003), gr-qc/0302072.

4 Parameters


amplitude	Scope: private	REAL

Description: Amplitude

Range		Default: 1.0
:


anl	Scope: private	REAL

Description: Amplitude of the non-linear Gaussian

Range		Default: 0.0
:


bound	Scope: private	STRING

Description: Boundary condition

Range		Default: static
.*	any registered boundary condition


compute_second_derivative_from_first_derivative	Scope: private	BOOLEAN

Description: Take first derivative twice to compute second derivate

		Default: no


deltanl	Scope: private	REAL

Description: sigma of the non-linear Gaussian

Range		Default: 0.0
:


eps	Scope: private	REAL

Description: A small number

Range		Default: 1.0e-10
0:*


epsx	Scope: private	REAL

Description: eps in x-direction of the non-linear Gaussian

Range		Default: 0.0
:


epsy	Scope: private	REAL

Description: eps in y-direction of the non-linear Gaussian

Range		Default: 0.0
:


initial_data	Scope: private	KEYWORD

Description: Type of initial data

Range		Default: plane
linear	x and y coordinates
plane	Plane wave
Gaussian	Gaussian wave packet
GaussianNonLinear	Gaussian wave packet for the non-linear RHS
GeneralMultipole	Multipole with arbitrary l and m
multipole	L=1 initial data, Gaussian in r
multipole l=1, m=0	L=1 m=0 initial data, Gaussian in r, u=0
multipole l=1, m=1	L=1 m=1 initial data, Gaussian in r, u=0
multipole l=1, m=-1	L=1 m=-1 initial data, Gaussian in r, u=0
multipole l=2	L=2 initial data, Gaussian in r
multipole l=2, u=0	L=2 initial data, Gaussian in r, u=0
multipole l=2, m=1	L=2 m=1 initial data, Gaussian in r, u=0
multipole l=2, m=-1	L=2 m=-1 initial data, Gaussian in r, u=0
multipole l=2, m=2	L=2 m=2 initial data, Gaussian in r, u=0
multipole l=2, m=-2	L=2 m=-2 initial data, Gaussian in r, u=0
multipole l=2, m=-2	L=2 m=-2 initial data, Gaussian in r, u=0
multipole l=4, m=0	L=4 m=0 initial data, Gaussian in r, u=0
multipole l=4, m=1	L=4 m=1 initial data, Gaussian in r, u=0
multipole l=4, m=-1	L=4 m=-1 initial data, Gaussian in r, u=0
multipole l=4, m=2	L=4 m=-2 initial data, Gaussian in r, u=0
multipole l=4, m=-2	L=4 m=-2 initial data, Gaussian in r, u=0
multipole l=4, m=3	L=4 m=3 initial data, Gaussian in r, u=0
multipole l=4, m=-3	L=4 m=-3 initial data, Gaussian in r, u=0
multipole l=4, m=4	L=4 m=4 initial data, Gaussian in r, u=0
multipole l=4, m=-4	L=4 m=-4 initial data, Gaussian in r, u=0
noise	Random noise
debug	number of current patch and grid point index


initial_data_analytic_derivatives	Scope: private	BOOLEAN

Description: Calculate spatial derivatives of the initial data analytically?

		Default: no


lapse	Scope: private	REAL

Description: Lapse function multiplier

Range		Default: 1.0
:	must not be zero


mass	Scope: private	REAL

Description: Mass M

Range		Default: 1.0
:


metric	Scope: private	KEYWORD

Description: Global metric

Range		Default: Minkowski
Minkowski	Minkowski
Kerr-Schild	Kerr-Schild
Kerr	Kerr Black Hole


multipole_l	Scope: private	INT

Description: For GeneralMultipole initial data: degree of spherical harmonic function

Range		Default: 2
0:*	A positive integer


multipole_m	Scope: private	INT

Description: For GeneralMultipole initial data: order of spherical harmonic function

Range		Default: 2
:	An integer -l<=m<=l


multipole_s	Scope: private	INT

Description: For GeneralMultipole initial data: spin weight spherical harmonic function

Range		Default: (none)
:	A positive integer


munl	Scope: private	REAL

Description: Speed of the non-linear Gaussian

Range		Default: 0.0
:


nonlinearrhs	Scope: private	BOOLEAN

Description: Add a non-linear term to the RHS?

		Default: no


numevolvedvars	Scope: private	INT

Description: The number of evolved variables in this thorn

Range		Default: 5
5:5	five


omenl	Scope: private	REAL

Description: Omega of the non-linear Gaussian

Range		Default: 0.0
:


outer_bound	Scope: private	STRING

Description: outer boundary

Range		Default: solution
zero	set all characteristics to zero
solution	use the same analytic solution as for the initial data
dirichlet	set all fields to zero
radiative	radiative boundary
none	no boundary condition


outer_penalty_bound	Scope: private	STRING

Description: outer penalty boundary

Range		Default: zero
zero	set all characteristics to zero
solution	use the same analytic solution as for the initial data


powerrhs	Scope: private	REAL

Description: Exponent in the non-linear RHS term

Range		Default: 5.0
3.0:13.0


radius	Scope: private	REAL

Description: Radius of the Gaussian

Range		Default: 0.0
0:*


recalculate_rhs	Scope: private	BOOLEAN

Description: Recalculate the RHSs in the ANALYSIS timebin

		Default: yes


rhsbound	Scope: private	STRING

Description: Boundary condition during RHS evaluation

Range		Default: none
.*	any registered boundary condition


rnl	Scope: private	REAL

Description: How fat the non-linear Gaussian is (r-R)

Range		Default: 0.0
:


shift	Scope: private	REAL

Description: Shift vector addition

Range		Default: 0.0
:


shift_interpolation_type	Scope: private	KEYWORD

Description: Setting for the interpolating vector field bî (only used for the db formulation)

Range		Default: shift
shift	Set bî = betaî (corresponds to d0 formulation)
zero	Set bî = 0 (corresponds to dk formulation)


shift_omega	Scope: private	REAL

Description: Rotational shift vector addition about z axis

Range		Default: 0.0
:


space_offset	Scope: private	REAL

Description: Space offset

Range		Default: 0.0
:


spin	Scope: private	REAL

Description: Spin a=J/M

Range		Default: 0.0
-1:+1


time_offset	Scope: private	REAL

Description: Time offset

Range		Default: 0.0
:


wave_number	Scope: private	REAL

Description: Wave number

Range		Default: 0.0
:


width	Scope: private	REAL

Description: Width of the Gaussian

Range		Default: 1.0
(0:*


mol_num_evolved_vars	Scope: shared from METHODOFLINES	INT

5 Interfaces

General

Implements:

llamawavetoy

Inherits:

grid

coordinates

globalderivative

summationbyparts

interpolate

Grid Variables

5.0.1 PRIVATE GROUPS


Group Names	Variable Names	Details

scalar		compact	0
	u	description	The scalar of the scalar wave equation fame
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar”
		timelevels	2
		variable type	REAL

density		compact	0
	rho	description	Time derivative of u
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar”
		timelevels	2
		variable type	REAL

dx_scalar		compact	0
	dx_u	description	Spatial derivatives of u
	dy_u	dimensions	3
	dz_u	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar”
		timelevels	2
		variable type	REAL

dxx_scalar		compact	0
	dxx_u	description	Spatial derivatives of u
	dyy_u	dimensions	3
	dzz_u	distribution	DEFAULT
	dxy_u	group type	GF
	dxz_u	tags	tensortypealias=”scalar”
	dyz_u	timelevels	2
		variable type	REAL

dx_density		compact	0
	dx_rho	description	Spatial derivatives of rho
	dy_rho	dimensions	3
	dz_rho	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar”
		timelevels	2
		variable type	REAL

velocity		compact	0
	vx	description	Spatial derivative of u
	vy	dimensions	3
	vz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar”
		timelevels	2
		variable type	REAL


Group Names	Variable Names	Details

debug		compact	0
	vxdebug	description	Spatial derivative of u
	vydebug	dimensions	3
	vzdebug	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar”
		timelevels	1
		variable type	REAL

scalardot		compact	0
	udot	description	RHS of u
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

densitydot		compact	0
	rhodot	description	RHS of rho
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

velocitydot		compact	0
	vxdot	description	RHS ov v
	vydot	dimensions	3
	vzdot	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

constraints		compact	0
	wx	description	Integrability condition
	wy	dimensions	3
	wz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” tensorparity=-1 Prolongation=”None”
		timelevels	1
		variable type	REAL

difference_v		compact	0
	diff_vx	description	Difference between v_i and d/dxî u
	diff_vy	dimensions	3
	diff_vz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL


Group Names	Variable Names	Details

velocity_squared		compact	0
	v2	description	Velocity squared
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

scalarenergy		compact	0
	energy	description	Energy of the scalar field
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

errors		compact	0
	error	description	Error of the solution
	error_rho	dimensions	3
	error_vx	distribution	DEFAULT
	error_vy	group type	GF
	error_vz	tags	tensortypealias=”scalar” Prolongation=”None”
	exact	timelevels	1
	exact_rho	variable type	REAL

errorsperiodic		compact	0
	errorperiodic	description	Error for a solution which is known to be periodic
	errorperiodic_rho	dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

metric		compact	0
	gxx	description	Spatial background metric
	gxy	dimensions	3
	gxz	distribution	DEFAULT
	gyy	group type	GF
	gyz	tags	tensortypealias=”scalar” Prolongation=”None”
	gzz	timelevels	1
		variable type	REAL

inverse_metric		compact	0
	guxx	description	Inverse of the spatial background metric
	guxy	dimensions	3
	guxz	distribution	DEFAULT
	guyy	group type	GF
	guyz	tags	tensortypealias=”scalar” Prolongation=”None”
	guzz	timelevels	1
		variable type	REAL


Group Names	Variable Names	Details

lapse		compact	0
	alpha	description	Spatial background metric
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

shift		compact	0
	betax	description	Spatial background metric
	betay	dimensions	3
	betaz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

volume_element		compact	0
	epsilon	description	Volume element due to the spatial background metric
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

min_spacing	min_spacing	compact	0
		description	Minimum grid spacing
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”None”
		timelevels	1
		variable type	REAL

6 Schedule

This section lists all the variables which are assigned storage by thorn Llama/LlamaWaveToy. 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:
scalar[2] density[2] velocity[2]
dx_scalar[2]
dxx_scalar[2]
dx_density[2]
errors
metric inverse_metric lapse shift volume_element
scalardot densitydot velocitydot

Scheduled Functions

CCTK_STARTUP

lwt_startup

	Language:	c
	Options:	meta
	Type:	function

MoL_Register (conditional)

lwt_register_mol

	Language:	c
	Options:	meta
	Type:	function

CCTK_ANALYSIS (conditional)

lwt_calc_rhs

calculate the rhs

	Language:	fortran
	Storage:	scalardot
		densitydot
		velocitydot
	Sync:	scalardot
		densitydot
		velocitydot
	Triggers:	scalardot
		densitydot
		velocitydot
	Type:	function

CCTK_ANALYSIS

lwt_calcenergy

calculate the energy of the scalar field

	Language:	fortran
	Storage:	scalarenergy
	Sync:	scalarenergy
	Triggers:	scalarenergy
	Type:	function

CCTK_ANALYSIS

lwt_error

calculate errors of the solution

	Language:	fortran
	Storage:	errorsperiodic
	Triggers:	errors
		errorsperiodic
	Type:	function

CCTK_ANALYSIS

lwt_min_spacing

calculate the smallest grid spacing

	Language:	fortran
	Storage:	min_spacing
	Sync:	min_spacing
	Triggers:	min_spacing
	Type:	function

CCTK_INITIAL

lwt_init_metric

initialise the metric

	Language:	fortran
	Type:	function

CCTK_INITIAL

lwt_calc_inverse_metric

transform the metric

	After:	lwt_init_metric
	Language:	fortran
	Type:	function

CCTK_INITIAL

lwt_init

initialise the system

	After:	lwt_init_metric
	Language:	fortran
	Type:	function

MoL_CalcRHS

lwt_calc_rhs

calculate the rhs

	Language:	fortran
	Type:	function

MoL_PostStep

lwt_outerboundary

apply outer boundaries

	Language:	fortran
	Type:	function

MoL_RHSBoundaries

lwt_rhs_outerboundary

apply mol rhs outer boundaries (eg. radiative boundary condition)

	Language:	fortran
	Type:	function

MoL_PostStep

lwt_boundaries

select the boundary condition

	After:	lwt_outerbound
	Language:	fortran
	Options:	level
	Sync:	scalar
		density
		velocity
	Type:	function

MoL_PostStep

applybcs

apply boundary conditions

	After:	lwt_boundaries
	Type:	group

Aliased Functions

Alias Name:	Function Name:
ApplyBCs	LWT_ApplyBCs