Using Cartoon2D

Denis Pollney

Date

Abstract

The Cartoon2D thorn allows fully 3D codes to be used to model axisymmetric systems, by considering a 3D evolution which is only one plane thick and applying a rotational tensor transformation to the flat faces of the plane as a boundary condition. This allows 3D codes to be tested efficiently on 2D problems, as well as providing a means of carrying out axisymmetric evolutions in cartesian coordinates without problems at the axis.

1 Background

The Cartoon2D thorn is the implementation of an idea first presented in [1]. The principle is to use a 3-dimensional Cartesian (x,y,z) coordinate grid which covers the y = 0 plane, but is only one finite difference molecule in width in the y-direction. The field variables in the y = 0 plane can be updated using standard 3D (x,y,z) coordinate finite differencing, with off-plane derivatives calculated by performing appropriate rotations of field variables using the axisymmetic assumption. This technique has the advantage of removing a number of the axis problems normally associated with axisymmetric codes.

The ‘cartoon’ method was first implemented in Cactus3 by Steve Brandt and Bernd Brügmann, and was translated to Cactus4 by Sai Iyer. Details of the method can be found in [1]. This document provides a practical guide to using the thorn as it is currently implemented.

2 Basic usage

Only a small number of parameters need to be set to use Cartoon2D.

cartoon_active
A flag which determines whether or not the cartoon boundary condition should be applied. This should be set to “yes” to use Cartoon2D, otherwise the evolution will be assumed to be standard 3D.
order
Determines the order of interpolations to be carried out. This will determine the number of ghost zones that you need to specify in the y-direction.
allow_grid_resize
When this flag is set to “yes”, Cartoon2D is allowed to modify the grid sizes specified in the parameter file so that a cartoon-compatible grid is used. See below.
verbose
Causes Cartoon2D to print a lot of messages.

For example, the following is a section of a parameter file which sets up a cartoon-style grid in bitant mode.

 
activethorns                            = "cartoon2d cartgrid3d pugh"  
 
cartoon2d::cartoon_active               = "yes"  
cartoon2d::order                        = 3  
cartoon2d::allow_grid_resize            = "yes"  
 
grid::type                              = "byspacing"  
grid::domain                            = "bitant"  
grid::bitant_plane                      = "xy"  
grid::dxyz                              = 0.2  
 
driver::global_nx                       = 16  
driver::global_ny                       = 3  
driver::global_nz                       = 16  
 
driver::ghost_size_x                    = 2  
driver::ghost_size_y                    = 1  
driver::ghost_size_z                    = 2  
 
driver::processor_topology              = "manual"  
driver::processor_topology_3d_x         = 1  
driver::processor_topology_3d_y         = 1  
driver::processor_topology_3d_z         = 2  
 
grid::avoid_originy                     = "no"  

The following features are worth noting:

Also, it is important to keep in mind that other thorns may also require their own parameters to be set in order to interact appropriately with Cartoon2D. For instance, see Section 4. Examples of working parameter files can be found in the Cartoon2D/test/ directory.


PIC

Figure 1: The cartoon plane layout. The symmetry axis is the z-axis, with grid functions defined on the y = 0 plane, in this example with two ghost zones.

3 Automatic grid re-sizing

The Cartoon2D thorn has some non-standard requirements of the grid which is normally set up by the grid implementation (for instance, by CartGrid3D). In particular,

These requirements (in particular the second) can not be met if the grid is specified byspacing (ie. by giving a dx value), since in this case the grid is set up assuming that the axes should be centred in each grid direction.

One way to get around this is to specify the grid byrange, giving minimum and maximum coordinate values for each axis so that the (dx,dy,dz) values are determined by dividing the range by the number of grid points. However, using this method it can be quite complicated to ensure that the grid spacing is even in each direction and that an appropriate number of ghost points extend past the z axis.

A simple hack to get around this complication is to specify the grid byspacing, but to set the allow_grid_resize parameter. In this case, the (dx,dy,dz) values can be set in the parameter file, and the grid ranges will be calculated based on the above criteria. Note that this involves modifying the “standard” behaviour of the CartGrid3D thorn (to place the z-axis along one edge of the grid). This is accomplished by internally resetting the grid type to byrange, and then specifying ranges which are compatible with the cartoon conditions and with the appropriate number of grid points and ghost zones. Note that this involves a reset of otherwise non-steerable parameters, and so must be scheduled at CCTK_RECOVER_PARAMETERS and before the grid is set up.

4 Interaction with other thorns

Some thorns will need to know that a cartoon grid is being used in order to function properly. In principle, they should use the cartoon_active parameter to check whether a cartoon grid is in use.

In practice, however, to avoid dependencies on Cartoon2D, it is often the case that the source code for such thorns make use of #ifdef statements to check whether Cartoon2D has been compiled in. Then, to check whether a cartoon grid is active, other thorn-specific parameters need to be set. For instance, if the ADM_BSSN thorn is being used for evolution, then the parameter

  adm_bssn::cartoon = "yes"

must be set. Similarly, the ADM evolution system requires adm::cartoon to be set. The AHFinder thorn requires that the parameter

  ahfinder::ahf_cartoon = "yes"

be set in order to use a cartoon grid.

Note that many thorns will require cartoon-specific modifications in order to be used with the Cartoon2D thorn. It should not be assumed that a given thorn will work with Cartoon2D unless it is specifically mentioned in the documentation or source code.

5 Source code

The interface to the Cartoon2D thorn is through the routines contained in the Cartoon2DBC.c source file. These routines can be used by other thorns (eg. evolution thorns) to apply cartoon boundary conditions to grid functions whenever it is appropriate to do so. The three interface functions are:

BndCartoon2DGN(cGH *GH, int tensortype, const char* group)
This function applies the cartoon boundary condition to the grid functions in the group specified by the group parameter. The tensortype argument parameter should be one of
TENSORTYPE_SCALAR
a scalar;
TENSORTYPE_U
a vector (single, upper index);
TENSORTYPE_DDSYM
a symmetric tensor with two lower indices.
BndCartoon2DVN(cGH *GH, int tensortype, const char *impvarname)
This function is like BndCartoon2DGN(), but operates on individual grid functions rather than groups.
BndCartoon2DVI(cGH *GH, int tensortype, int vi)
This function is like BndCartoon2DGN(), but operates takes a grid function index to indicate the grid function to which the condition should be applied.