Boundary Conditions

Date

Abstract

Provides a generic interface to boundary conditions, and provides a set of standard boundary conditions for one, two, and three dimensional grid variables.

1 Introduction

This thorn provides a generic method for registering routines to perform boundary conditions, and selecting variables to have these boundary conditions applied to them. In addition, it provides abstraction which allows all considerations of symmetry to be separated from those of physical boundary conditions. The general idea is that codes which use boundary conditions, be they physical or symmetry conditions, need not know anything about the thorns which provide them.

This thorn also contains some standard boundary conditions, most of which can be used with any spatial dimension and data type.

1.1 Local and non-local boundary conditions

Boundary conditions can be local, meaning that the boundary point can be updated based on data in its immediate vicinity, or non-local, meaning that the new value on the boundary depends on data from a remote region of the computational domain (for a parallel simulation this data could for example be physically located on several diﬀerent processors). An example of the latter is a “rotating” symmetry condition, which arises e.g. when one uses a quadrant to simulate a physical domain which possesses a rotational symmetry.

1.2 Symmetry and physical boundary conditions

Symmetry boundary conditions are those that arise by viewing the computational domain as a subregion of some larger domain which possesses symmetries. These symmetries allow a simulation of the subregion to act as an eﬀective simulation of the larger encompassing domain, because the latter can be inferred from the former via the symmetry. For example, one can often simulate a rotating star by ‘slicing’ the space in half through the equatorial plane, simulating only one half, and placing a reﬂection boundary condition at this plane. The symmetry can be regarded as a property of the underlying computational grid upon which the simulation takes place.

It is often possible to describe the symmetry of a physical problem in terms of multiple ‘simpler’ symmetries. Going back to the rotating star example, we can note that there is also a rotational symmetry about the axis of the star. Thus it is suﬃcient to simulate only the upper half of a $\varphi =\text{const}$ plane of the star, since rotational symmetry will recover half of the star from the single plane, and the reﬂection symmetry can recover the other half of the star. To do this we use two symmetry boundary conditions, one for the rotational symmetry and one for the reﬂection symmetry. At the edges and corner grid points there will be two symmetry boundary conditions active, which illustrates a general point about symmetry boundary conditions, namely that there can be any number of them active at any given grid point. In addition symmetry boundary conditions are often non-local, for example a periodic boundary condition which applies in simulating plasma in a tokamak.

Physical boundary conditions are motivated by the physics of the quantity that the grid variable represents, such as one which allows outgoing waves of a scalar ﬁeld to propagate oﬀ the grid, but does not allow ingoing waves or reﬂections. Usually the same physical boundary condition is applied to all external boundaries of the computational domain, however this is not always the case. Currently thorn Boundary allows a separate boundary condition to be applied to each face of the domain, however this is only implemented at the moment using the older deprecated interface. Face speciﬁc calls will be available using the current interface shortly. It is also possible that one will want to use diﬀerent physical boundary conditions at diﬀerent regions of a face, and support for this can be added if necessary. Usually physical boundary conditions are local. A non-local physical boundary condition may arise e.g. from a need to solve an elliptic equation at the boundary. As opposed to symmetry boundary conditions, it only makes sense to have a single physical boundary condition active at a given grid point.1

To summarize, a ‘physics’ thorn, such as a spacetime evolution thorn, knows only about physical boundary conditions. Symmetry boundary conditions are aspects of the grid and are managed by other thorns.

2 The generic boundary interface

The implementation Boundary provides a number of aliased functions, which allow application thorns to register routines which provide a particular physical boundary condition, and also to select variables or groups of variables to have boundary conditions applied to whenever the ApplyBCs schedule group is scheduled (see section 2.3). In addition, an aliased function is provided which returns a list of grid variables selected for a boundary condition (see appendix 11).

2.1 Boundary condition registration

To register a routine to provide some physical boundary condition, use

Boundary_RegisterPhysicalBC(CCTK_POINTER cctkGH,
phys_bc_fn_ptr function_pointer,
CCTK_STRING bc_name)

where

 cctkGH pointer to the grid hierarchy

 function_pointer pointer to the function providing the boundary condition

 bc_name name of boundary condition

The function pointer takes arguments

(CCTK_POINTER cctkGH, CCTK_INT num_vars, CCTK_INT *var_indices,
CCTK_INT *faces, CCTK_INT *widths, CCTK_INT *table_handles)

(this deﬁnes the type phys_bc_fn_ptr, above) where

 cctkGH pointer to the grid hierarchy

 num_vars number of entries passed in the following three arrays

 var_indices array of num_vars variable indices selected for this boundary condition

 faces array of num_vars faces speciﬁcations (see section 2.4)

 widths array of num_vars boundary widths (see below)

 table_handles array of num_vars table handles

The table handles hold extra arguments for each application of the boundary condition. The four arrays are sorted ﬁrst on table handle, and then on variable index. This way variables which have precisely the same boundary condition are grouped together, and within this grouping variables are sorted by index, so that variable groups are stored together. In many cases this sorting will allow a more eﬃcient implementation of the boundary condition. (At the moment it is not clear how face information should be considered in the sorting, so it is not.) A null pointer can be passed for function_pointer, in which case no routine is executed when Boundary_ApplyPhysicalBCs is called (see section 2.3).

2.2 Boundary condition selection

To select a grid variable to have a boundary condition applied to it, use one of the following aliased functions:

Boundary_SelectVarForBC(CCTK_POINTER cctkGH,
CCTK_INT faces,
CCTK_INT boundary_width,
CCTK_INT table_handle,
CCTK_STRING var_name,
CCTK_STRING bc_name)

Boundary_SelectVarForBCI(CCTK_POINTER cctkGH,
CCTK_INT faces,
CCTK_INT boundary_width,
CCTK_INT table_handle,
CCTK_INT var_index,
CCTK_STRING bc_name)

Boundary_SelectGroupForBC(CCTK_POINTER cctkGH,
CCTK_INT faces,
CCTK_INT boundary_width,
CCTK_INT table_handle,
CCTK_STRING group_name,
CCTK_STRING bc_name)

Boundary_SelectGroupForBCI(CCTK_POINTER cctkGH,
CCTK_INT faces,
CCTK_INT boundary_width,
CCTK_INT table_handle,
CCTK_INT group_index,
CCTK_STRING bc_name)

where

 cctkGH pointer to the grid hierarchy

 faces set of faces to which to apply the boundary condition

 boundary_width width (in grid points) of the boundaries

 table_handle handle for table which holds extra arguments for the boundary condition

 var_name name of the grid variable

 bc_name name of the boundary condition

 var_index index of grid variable

 group_name name of group of grid variables

 group_index index of group of grid variables

Each of these functions returns 0 for success, or a negative error code if something went wrong.

Boundary_SelectVarForBC() and Boundary_SelectVarForBCI() select a single grid variable for a boundary condition, using either the variable name or index respectively. Boundary_SelectGroupForBC() and Boundary_SelectGroupForBCI() select an entire variable group, using either its name or index.

Each of these functions takes a faces speciﬁcation, a boundary width, and a table handle as additional arguments. The faces speciﬁcation is a single integer which identiﬁes a set of faces to which to apply the boundary condition. See section 2.4 for details. The boundary width is the thickness, in grid points, of the boundaries.

The table handle identiﬁes a table which holds extra arguments for the particular boundary condition that is requested. For example, if a negative value is passed for the boundary width, then the boundary condition will look in this table for a $2d$-element integer array, which holds the width of each face of the boundary (for a $d$ dimensional grid variable). (The ﬁrst element of the array holds the width of the ‘-x’ face, the second the ‘+x’ face, the third the ‘-y’ face, etc.)

In some cases the table handle is required, so the boundary condition, when it is called within the BoundaryConditions schedule group (see section 2.3), will return an error code. However, in most cases it is optional. If one uses an invalid table handle here (such as -1), commonly used default values will be assumed for all arguments (besides the explicit faces speciﬁcation and boundary widths). Note that you, the user, will be creating the table, so you may choose whatever options (such as case sensitivity) you like. The case of the keys for which the boundary conditions implemented in this thorn search are as given in the documentation, which is currently all capitals. To be safe you may choose to create case-insensitive tables, however case sensitive tables are slightly faster.

The name of the boundary condition must match that with which the boundary condition providing function was registered. These names are case insensitive. See section 3 for a list of boundary conditions provided by thorn Boundary.

2.3 Schedule groups

Implementation Boundary creates two schedule groups

ApplyBCs

and

BoundaryConditions in ApplyBCs BEFORE Boundary_ClearSelection

and schedules two functions Boundary_ApplyPhysicalBCs in BoundaryConditions and Boundary_ClearSelection in ApplyBCs. Boundary_ApplyPhysicalBCs goes through the list of all selected grid variables, and calls the registered function corresponding to the requested boundary condition on each. Boundary_ClearSelection clears the list of selected grid variables. A thorn which wishes to have boundary conditions applied simply schedules ApplyBCs at the appropriate point. Please schedule it as e.g. <MyThorn>_ApplyBCs, to make each instance of it unique. Boundary_ClearSelection ensures that each boundary condition gets executed exactly once for each selected grid variable.

2.4 Faces

The computational domain is assumed to be in the shape of a $n$-dimensional ‘rectangle’, which has $2n$ $n-1$ dimensional faces. (Usually $n$ is three.) Each of these $2n$ faces is assigned a speciﬁc bit in a word, so that arbitrary subsets can be compactly expressed as a bitwise-or of these individual bits. Macros deﬁning this mapping of subsets to bits will be provided. For the moment there is only CCTK_ALL_FACES, which corresponds to the set of all faces of the domain. If you need face speciﬁc calls immediately, please use the old interface for now.

The mapping of bits to faces will likely be the same as that used for the (optional) BOUNDARY_WIDTH array. Precisely, the rule is as follows. For a $d$ dimensional grid variable, label the elements or bits by integers $i$ from $0$ to $2d-1$. Element or bit $i$ gets mapped to face ${\left(-\right)}^{i+1}{e}_{⌊i⌋}$, where ${}_{⌊⌋}$ designates the ‘ﬂoor’ function (greatest integer less than its argument), and ${e}_{j}$ represents the ‘$j$-direction’ on the grid.

3 Provided boundary conditions

Thorn Boundary also provides seven standard boundary conditions, which can be applied to one, two, or three dimensional grid variables. The boundary conditions available are

• Scalar
• Flat
• Copy
• Robin
• Static
• None

Registration for each of these can be switched oﬀ by setting any of the following parameters to “no” (each defaults to “yes”)

• register_scalar
• register_flat
• register_copy
• register_robin
• register_static
• register_none

This is useful if you have your own implementation of one of these boundary conditions, which you would like to use instead.

Note that the number of boundary zones, as expressed in the boundary_width argument or the BOUNDARY_WIDTH array, is taken from the total number of grid points presented by Cactus through cctk_lsh, etc.

For the moment, these boundary routines may not work properly on grid arrays. Please contact
cactusmaint@cactuscode.org if you run into trouble.

3.1.1 Old interface

The old, direct function call interface to these boundary conditions is still available, and is documented here, though it is deprecated and will be removed at some point in the future. It is provided for a number of reasons, the most signiﬁcant of which is to provide compatibility with existing codes. Another reason why you might choose to use the old interface is if you have diﬃculty doing your iterations with the Cactus scheduler, and thus have trouble scheduling the ApplyBCs schedule group everywhere you need boundary conditions applied. A third reason may be that you need face speciﬁc calls immediately.

You should not run into any special diﬃculty mixing the old and new interface, just be aware of the order in which boundary conditions, and code that depends upon them, are executed.

Note that if you choose to use the old interface for some boundary conditions, then the symmetry conditions will not know to apply themselves to those grid variables for which you use the old interface. To get around this diﬃculty, simply select these grid variables for the “None” boundary condition, and be sure that ApplyBCs is scheduled in an appropriate place.

All routines can be called by

• variable name: (<implementation>:<var_name> ) Suﬃx: VN; apply the boundary condition to the variable with the speciﬁed name.
• group name: (<implementation>:<group_name>) Suﬃx: GN; apply the boundary condition to all variables in the group.
• variable index: Suﬃx: VI; apply the boundary condition to the variable with the speciﬁed variable index.
• group index: Suﬃx: GI apply the boundary condition to all variables in the group with the speciﬁed group index.

For the boundary conditions in individual coordinate directions, use

 dir=-1 to apply at $x={x}_{min}$ dir= 1 to apply at $x={x}_{max}$ dir=-2 to apply at $y={y}_{min}$ dir= 2 to apply at $y={y}_{max}$ dir=-3 to apply at $z={z}_{min}$ dir= 3 to apply at $z={z}_{max}$

Prototypes for each of the functions described for the old interface are included in the header ﬁle Boundary.h. Please add

uses include header: Boundary.h

to your thorn’s interface.ccl to use this header ﬁle in your C/C++ source ﬁles.

4 Scalar Boundary Condition

A scalar boundary condition means that the value of the given ﬁeld or ﬁelds at the boundary is set to a given scalar value, for example zero. The scalar boundary condition is registered under the name “Scalar”.

A table passed to the scalar boundary condition may contain the following additional arguments:

 key variable type description default value SCALAR CCTK_REAL the scalar value 0.0 BOUNDARY_WIDTH CCTK_INT array stencil width for each face n/a

4.2 Old interface

Calling from C:

All Coordinate Directions:

int ierr = BndScalarVN(cGH *cctkGH, int *stencil_size,
CCTK_REAL var0, char *variable_name)
int ierr = BndScalarGN(cGH *cctkGH, int *stencil_size,
CCTK_REAL var0, char *group_name)
int ierr = BndScalarVI(cGH *cctkGH, int *stencil_size,
CCTK_REAL var0, int group_index)
int ierr = BndScalarGI(cGH *cctkGH, int *stencil_size,
CCTK_REAL var0, int variable_index)

Individual Coordinate Directions:

int ierr = BndScalarDirVN(cGH *cctkGH, int stencil, int dir,
CCTK_REAL var0, char *variable_name)
int ierr = BndScalarDirGN(cGH *cctkGH, int stencil, int dir,
CCTK_REAL var0, char *group_name)
int ierr = BndScalarDirVI(cGH *cctkGH, int stencil, int dir,
CCTK_REAL var0, int group_index)
int ierr = BndScalarDirGI(cGH *cctkGH, int stencil, int dir,
CCTK_REAL var0, int variable_index)

Calling from Fortran:

All Coordinate Directions:

call BndScalarVN(ierr, cctkGH, stencil_size, var0, variable_name)
call BndScalarGN(ierr, cctkGH, stencil_size, var0, group_name)
call BndScalarVI(ierr, cctkGH, stencil_size, var0, variable_index)
call BndScalarGI(ierr, cctkGH, stencil_size, var0, group_index)

Individual Coordinate Directions:

call BndScalarDirVN(ierr, cctkGH, stencil, dir, var0, variable_name)
call BndScalarDirGN(ierr, cctkGH, stencil, dir, var0, group_name)
call BndScalarDirVI(ierr, cctkGH, stencil, dir, var0, variable_index)
call BndScalarDirGI(ierr, cctkGH, stencil, dir, var0, group_index)

where

 integer ierr

 CCTK_POINTER cctkGH

 integer dir

 integer stencil

 integer stencil_size(dim)

 CCTK_REAL var0

 character*(*) variable_name

 character*(*) group_name

 integer variable_index

 integer group_index

Arguments
 ierr Return value, negative value indicates the boundary condition was not successfully applied

 cctkGH Grid hierarchy pointer

 var0 Scalar value to apply (For a complex grid function, this is the real part,

 the imaginary part is set to zero.)

 dir Coordinate direction in which to apply boundary condition

 stencil_size Array with dimension of the grid function, containing the stencil width

 variable_name Name of the variable

 group_name Name of the group

 variable_index Variable index

 group_index Group index

5 Flat Boundary Condition

A ﬂat boundary condition means that the value of the given ﬁeld or ﬁelds at the boundary is copied from the value one grid point in, in any direction. For example, for a stencil width of one, the boundary value of phi phi(nx,j,k), on the positive x-boundary will be copied from phi(nx-1,j,k). The ﬂat boundary condition is registered under the name “Flat”.

A table passed to the ﬂat boundary condition may contain the following additional arguments:

 key variable type description default value BOUNDARY_WIDTH CCTK_INT array stencil width for each face n/a

5.2 Old interface

Calling from C:

All Coordinate Directions:

int ierr = BndFlatVN(cGH *cctkGH, int *stencil_size, char *variable_name)
int ierr = BndFlatGN(cGH *cctkGH, int *stencil_size, char *group_name)
int ierr = BndFlatVI(cGH *cctkGH, int *stencil_size, int variable_index)
int ierr = BndFlatGI(cGH *cctkGH, int *stencil_size, int group_index)

Individual Coordinate Directions:

int ierr = BndFlatDirVN(cGH *cctkGH, int stencil, int dir, char *variable_name)
int ierr = BndFlatDirGN(cGH *cctkGH, int stencil, int dir, char *group_name)
int ierr = BndFlatDirVI(cGH *cctkGH, int stencil, int dir, int variable_index)
int ierr = BndFlatDirGI(cGH *cctkGH, int stencil, int dir, int group_index)

Calling from Fortran:

All Coordinate Directions:

call BndFlatVN(ierr, cctkGH, stencil_array, variable_name)
call BndFlatGN(ierr, cctkGH, stencil_array, group_name)
call BndFlatVI(ierr, cctkGH, stencil_array, variable_index)
call BndFlatGI(ierr, cctkGH, stencil_array, group_index)

Individual Coordinate Directions:

call BndFlatDirVN(ierr, cctkGH, stencil, dir, variable_name)
call BndFlatDirGN(ierr, cctkGH, stencil, dir, group_name)
call BndFlatDirVI(ierr, cctkGH, stencil, dir, variable_index)
call BndFlatDirGI(ierr, cctkGH, stencil, dir, group_index)

where

 integer ierr

 CCTK_POINTER cctkGH

 integer dir

 integer stencil

 integer stencil_array(dim)

 character*(*) variable_name

 character*(*) group_name

 integer variable_index

 integer group_index

Arguments
 ierr Return value, negative value indicates the boundary condition was not successfully applied

 cctkGH Grid hierarchy pointer

 dir Coordinate direction in which to apply boundary condition

 stencil_size Array with dimension of the grid function, containing the stencil width

 variable_name Name of the variable

 group_name Name of the group

 variable_index Variable index

 group_index Group index

6 Radiation Boundary Condition

This is a two level scheme. Grid functions are given for the current time level (to which the BC is applied) as well as grid functions from a past timelevel which are needed for constructing the boundary condition. The grid function of the past time level needs to have the same geometry. Currently radiative boundary conditions can only be applied with a stencil width of one in each direction.

The radiative boundary condition that is implemented is

 $f={f}_{0}+\frac{u\left(r-vt\right)}{r}+\frac{h\left(r+vt\right)}{r}$ (1)

That is, outgoing radial waves with a $1∕r$ fall oﬀ, and the correct asymptotic value ${f}_{0}$ are assumed, including the possibility of incoming waves (these incoming waves should be modeled somehow).

Condition 1 above leads to the diﬀerential equation:

 $\frac{{x}^{i}}{r}\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial {x}^{i}}+\frac{v{x}^{i}}{{r}^{2}}\left(f-{f}_{0}\right)=H\frac{v{x}^{i}}{{r}^{2}}$ (2)

where ${x}^{i}$ is the normal direction to the given boundaries, and $H=2dh\left(s\right)∕ds$.

At a given boundary only the derivatives in the normal direction are considered. Notice that $u\left(r-vt\right)$ has disappeared, but we still do not know the value of $H$.

To get $H$ we do the following: The expression is evaluated one point in from the boundary and solved for $H$ there. Now we need a way of extrapolating $H$ to the boundary. For this, assume that $H$ falls oﬀ as a power law:

 (3)

The value of $n$ is deﬁned by the parameter radpower. If this parameter is negative, $H$ is forced to be zero (this corresponds to pure outgoing waves and is the default).

The observed behavior is the following: Using $H=0$ is very stable, but has a very bad initial transient. Taking $n$ to be 0 or positive improves the initial behavior considerably, but introduces a drift that can kill an evolution at very late times. Empirically, the best value found so far is $n=2$, for which the initial behavior is very nice, and the late time drift is quite small.

Another problem with this condition is that it does not use the physical characteristic speed, but rather it assumes a wave speed of $v$, so the boundaries should be out in the region where the characteristic speed is constant. Notice that this speed does not have to be 1.

The radiation boundary condition is registered under the name “Radiation”.

A table passed to the radiative boundary condition may contain the following additional arguments:

 key variable type description default value LIMIT CCTK_REAL ${f}_{0}$ 0.0 PREVIOUS_TIME_LEVEL CCTK_INT or CCTK_STRING GV which holds the Cactus previous time level previous time level SPEED CCTK_REAL wave speed $v$ 1.0 BOUNDARY_WIDTH CCTK_INT array stencil width for each face n/a

The default behavior is to use the Cactus previous time level, as deﬁned in the interface.ccl ﬁle, for the grid variable requested for the radiative boundary condition. The “PREVIOUS_TIME_LEVEL” key is provided for backward compatibility only, and will be deprecated in the future. The corresponding value may be either a CCTK_INT, which will be interpreted as the index of a grid variable holding the previous time level, or a CCTK_STRING, interpreted as holding the name. Note that this will not work when selecting an entire variable group (with more than one member) with one call to Boundary_SelectGroupForBC*, as each member will have a separate previous time level, and thus require a separate table. Please make your life easier by using Cactus time levels…

6.2 Old interface

Calling from C:

All Coordinate Directions:

int ierr = BndRadiativeVN(cGH *cctkGH, int *stencil_size,
CCTK_REAL limit, CCTK_REAL speed,
char *variable_name, char *variable_name_past)
int ierr = BndRadiativeGN(cGH *cctkGH, int *stencil_size,
CCTK_REAL limit, CCTK_REAL speed,
char *group_name, char *group_name_past)
int ierr = BndRadiativeVI(cGH *cctkGH, int *stencil_size,
CCTK_REAL limit, CCTK_REAL speed,
int variable_index, int variable_index_past)
int ierr = BndRadiativeGI(cGH *cctkGH, int *stencil_size,
CCTK_REAL limit, CCTK_REAL speed,
int group_index, int group_index_past)

Individual Coordinate Directions:

int ierr = BndRadiativeDirVN(cGH *cctkGH, int stencil, int dir,
CCTK_REAL limit, CCTK_REAL speed,
char *variable_name, char *variable_name_past)
int ierr = BndRadiativeDirGN(cGH *cctkGH, int *stencil, int dir,
CCTK_REAL limit, CCTK_REAL speed,
char *group_name, char *group_name_past)
int ierr = BndRadiativeDirVI(cGH *cctkGH, int *stencil, int dir,
CCTK_REAL limit, CCTK_REAL speed,
int variable_index, int variable_index_past)
int ierr = BndRadiativeDirGI(cGH *cctkGH, int *stencil, int dir,
CCTK_REAL limit, CCTK_REAL speed,
int group_index, int group_index_past)

Calling from Fortran:

All Coordinate Directions:

call BndRadiativeVN(ierr, cctkGH, stencil_size, speed, limit,
variable_name, variable_name_past)
call BndRadiativeGN(ierr, cctkGH, stencil_size, speed, limit,
group_name, group_name_past)
call BndRadiativeVI(ierr, cctkGH, stencil_size, speed, limit,
variable_index, variable_index_past)
call BndRadiativeGI(ierr, cctkGH, stencil_size, speed, limit,
group_index, group_index_past)

Individual Coordinate Directions:

call BndRadiativeDirVN(ierr, cctkGH, stencil, dir, speed, limit,
variable_name, variable_name_past)
call BndRadiativeDirGN(ierr, cctkGH, stencil, dir, speed, limit,
group_name, group_name_past)
call BndRadiativeDirVI(ierr, cctkGH, stencil, dir, speed, limit,
variable_index, variable_index_past)
call BndRadiativeDirGI(ierr, cctkGH, stencil, dir, speed, limit,
group_index, group_index_past)

where

 integer ierr

 CCTK_POINTER cctkGH

 integer dir

 integer stencil

 integer stencil_array(dim)

 character*(*) variable_name

 character*(*) group_name

 integer variable_index

 integer group_index

 CCTK_REAL speed

 CCTK_REAL limit

Arguments
 ierr return value, operation failed when return value negative

 cctkGH grid hierarchy pointer

 stencil_size(dim) array of size dim (dimension of the grid function).

 To how many points from the outer boundary to apply the boundary condition.

 speed wave speed used for boundary condition ($v$)

 limit asymptotic value of function at inﬁnity (${f}_{0}$)

 variable_name the name of the grid function to which the boundary condition will be applied

 variable_name_past The name of the grid function containing the values on the past time level,

 needed to calculate the boundary condition.

 group_name the name of the group to which the boundary condition will be applied

 group_name_past is the name of the group containing the grid functions on the past time level,

 needed to calculate the boundary condition.

 variable_index the index of the grid function to which the boundary condition will be applied

 variable_index_past the index of the grid function containing the values on the past time level,

 needed to calculate the boundary condition.

 group_index the index of the group to which the boundary condition will be applied

 group_index_past the index of the group containing the values on the past time level,

 needed to calculate the boundary condition.

7 Copy Boundary Condition

This is a two level scheme. Copy the boundary values from a diﬀerent grid function, for example the previous timelevel. The two grid functions (or groups of grid functions) must have the same geometry. The copy boundary condition is registered under the name “Copy”.

The “COPY_FROM” argument for the copy boundary condition is required, so a valid table handle is required as well. The keys read are

 key variable type description default value COPY_FROM CCTK_INT or CCTK_STRING GV to copy from no default BOUNDARY_WIDTH CCTK_INT array stencil width for each face n/a

(The BOUNDARY_WIDTH table entry is only necessary if the boundary_width parameter is negative.)

7.2 Old interface

Calling from C:
int ierr = BndCopyVN(cGH *cctkGH, int *stencil_size,
char *variable_name_to, char *variable_name_from)
int ierr = BndCopyGN(cGH *cctkGH, int *stencil_size,
char *group_name_to, char *group_name_from)
int ierr = BndCopyVI(cGH *cctkGH, int *stencil_size,
int variable_index_to, int variable_index_from)
int ierr = BndCopyGI(cGH *cctkGH, int *stencil_size,
int group_index_to, int group_index_from)

Calling from Fortran:
call BndCopyVN(ierr, cctkGH, stencil_size, variable_name_to,
variable_name_from)
call BndCopyVN(ierr, cctkGH, stencil_size, group_name_to,
group_name_from)
call BndCopyVN(ierr, cctkGH, stencil_size, variable_index_to,
variable_index_from)
call BndCopyVN(ierr, cctkGH, stencil_size, group_index_to,
group_index_from)

where

 integer ierr return value, operation failed when return value negative

 CCTK_POINTER cctkGH grid hierarchy pointer

 integer stencil_size(dim) array of size dim (dimension of the grid function). To how many points

 from the outer boundary to apply the boundary condition.

 character*(*) variable_name_to the name of the grid function to which the boundary condition

 will be applied by copying to.

 character*(*) variable_name_from the name of the grid function containing the values to copy from.

 character*(*) group_name_to the name of the group to which the boundary condition

 will be applied by copying to.

 character*(*) group_name_from the name of the group containing the the values to copy from.

 integer variable_index_to the index of the grid function to which the boundary condition

 will be applied by copying to.

 integer variable_index_from the index of the grid function containing the the values to copy from.

 integer group_index_to the index of the group to which the boundary condition

 will be applied by copying to.

 integer group_index_from the index of the group containing the the values to copy from.

8 Robin Boundary Condition

This boundary condition has not yet been implemented in individual coordinate directions. The Robin boundary condition is:

 $f\left(r\right)={f}_{0}+\frac{k}{{r}^{n}}$ (4)

with $k$ a constant, $n$ the decay rate and ${f}_{0}$ the value at inﬁnity. This implies:

 $\frac{\partial f}{\partial r}=-n\frac{k}{{r}^{n+1}}$ (5)

or

 $\frac{\partial f}{\partial r}=-n\frac{\left(f-{f}_{0}\right)}{r}$ (6)

Considering now a given Cartesian direction $x$ we get:

 $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial r}\frac{\partial r}{\partial x}=\frac{x}{r}\frac{\partial f}{\partial r}$ (7)

which implies

 $\frac{\partial f}{\partial x}=-n\left(f-{f}_{0}\right)\frac{x}{{r}^{2}}$ (8)

The equations are then ﬁnite diﬀerenced around the grid point $i+1∕2$:

 ${f}_{i+1}-{f}_{i}=-n\Delta x\left(\frac{1}{2}\left({f}_{i+1}+{f}_{i}\right)-{f}_{0}\right)\frac{{x}_{i+1∕2}}{{r}_{i+1∕2}^{2}}$ (9)

or

 ${f}_{i+1}-{f}_{i}=-n\Delta x\left(\left({f}_{i+1}+{f}_{i}\right)-2{f}_{0}\right)\frac{{x}_{i+1}+{x}_{i}}{{\left({r}_{i+1}+{r}_{i}\right)}^{2}}$ (10)

And this is then solved either for ${f}_{i}$ or ${f}_{i+1}$ depending on which side are we looking at.

The Robin boundary condition is registered under the name “Robin”.

A table passed to the Robin boundary condition may contain the following additional arguments:

 key variable type description default value FINF CCTK_REAL ${f}_{0}$ 0 DECAY_POWER CCTK_INT $n$ 1 BOUNDARY_WIDTH CCTK_INT array stencil width for each face n/a

8.2 Old interface

Calling from C:

All Coordinate Directions:

int ierr = BndRobinVN(cGH *cctkGH, int *stencil_size,
CCTK_REAL finf, int npow, char *variable_name)
int ierr = BndScalarGN(cGH *cctkGH, int *stencil_size,
CCTK_REAL finf, int npow, char *group_name)
int ierr = BndScalarVI(cGH *cctkGH, int *stencil_size,
CCTK_REAL finf, int npow, int group_index)
int ierr = BndScalarGI(cGH *cctkGH, int *stencil_size,
CCTK_REAL finf, int npow, int variable_index)

Calling from Fortran:

All Coordinate Directions:

call BndRobinVN(ierr, cctkGH, stencil_size, finf, npow, variable_name)
call BndRobinGN(ierr, cctkGH, stencil_size, finf, npow, group_name)
call BndRobinVI(ierr, cctkGH, stencil_size, finf, npow, variable_index)
call BndRobinGI(ierr, cctkGH, stencil_size, finf, npow, group_index)

where

 integer ierr

 CCTK_POINTER cctkGH

 integer stencil_size(dim)

 CCTK_REAL finf

 integer npow

 character*(*) variable_name

 character*(*) group_name

 integer variable_index

 integer group_index

Arguments
 ierr Return value, negative value indicates the boundary condition was not successfully applied

 cctkGH Grid hierarchy pointer

 finf Scalar value at inﬁnity

 npow Decay rate ($n$ in discussion above)

 stencil_size Array with dimension of the grid function, containing the stencil width to apply the boundary at

 variable_name Name of the variable

 group_name Name of the group

 variable_index Variable index

 group_index Group index

9 Static Boundary Condition

The static boundary condition ensures that the boundary values do not evolve in time, by copying their values from previous timelevels. The static boundary condition is registered under the name “Static”.

A table passed to the static boundary condition may contain the following additional arguments:

 key variable type description default value BOUNDARY_WIDTH CCTK_INT array stencil width for each face n/a

9.2 Old interface

Calling from C:

All Coordinate Directions:

int ierr = BndStaticVN(cGH *cctkGH, int *stencil_size, char *variable_name)
int ierr = BndStaticGN(cGH *cctkGH, int *stencil_size, char *group_name)
int ierr = BndStaticVI(cGH *cctkGH, int *stencil_size, int variable_index)
int ierr = BndStaticGI(cGH *cctkGH, int *stencil_size, int group_index)

Individual Coordinate Directions:

int ierr = BndStaticDirVN(cGH *cctkGH, int stencil, int dir, char *variable_name)
int ierr = BndStaticDirGN(cGH *cctkGH, int stencil, int dir, char *group_name)
int ierr = BndStaticDirVI(cGH *cctkGH, int stencil, int dir, int variable_index)
int ierr = BndStaticDirGI(cGH *cctkGH, int stencil, int dir, int group_index)

Calling from Fortran:

All Coordinate Directions:

call BndStaticVN(ierr, cctkGH, stencil_array, variable_name)
call BndStaticGN(ierr, cctkGH, stencil_array, group_name)
call BndStaticVI(ierr, cctkGH, stencil_array, variable_index)
call BndStaticGI(ierr, cctkGH, stencil_array, group_index)

Individual Coordinate Directions:

call BndStaticDirVN(ierr, cctkGH, stencil, dir, variable_name)
call BndStaticDirGN(ierr, cctkGH, stencil, dir, group_name)
call BndStaticDirVI(ierr, cctkGH, stencil, dir, variable_index)
call BndStaticDirGI(ierr, cctkGH, stencil, dir, group_index)

where

 integer ierr

 CCTK_POINTER cctkGH

 integer dir

 integer stencil

 integer stencil_array(dim)

 character*(*) variable_name

 character*(*) group_name

 integer variable_index

 integer group_index

Arguments
 ierr Return value, negative value indicates the boundary condition was not successfully applied

 cctkGH Grid hierarchy pointer

 dir Coordinate direction in which to apply boundary condition

 stencil_size Array with dimension of the grid function, containing the stencil width to apply the boundary at

 variable_name Name of the variable

 group_name Name of the group

 variable_index Variable index

 group_index Group index

10 None Boundary Condition

The “None” boundary condition does just that, nothing. It is provided to inform the boundary implementation of grid variables which should have symmetry boundary conditions applied to them, but do not have their physical boundary conditions applied using a properly registered function.