## A Multi-Patch Wave Toy

June 22, 2024

### 1 Physical System

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

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

and the shift speed modes are $$d_A$$, with $$A$$ transversal directions.

The physical energy is

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

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

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

register banner with cactus

 Language: c Options: meta Type: function

MoL_Register (conditional)

lwt_register_mol

register variables with 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