/[MITgcm]/manual/s_phys_pkgs/text/exch2.tex

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-revision 1.11 by afe,
Mon Mar 15 22:39:28 2004 UTC
+revision 1.13 by afe,
Wed Mar 17 19:49:22 2004 UTC
 Line 16
  \subsection{Introduction}
- The \texttt{exch2} package is an extension to the original cubed
+ The \texttt{exch2} package extends the original cubed
- sphere topological configuration that allows more flexible domain
+ sphere topology configuration to allow more flexible domain
  decomposition and parallelization.  Cube faces (also called
  subdomains) may be divided into any number of tiles that divide evenly
  into the grid point dimensions of the subdomain.  Furthermore, the
- individual tiles may be run on separate processors in different
+ individual tiles can run on separate processors in different
  combinations, and whether exchanges between particular tiles occur
  between different processors is determined at runtime.  This
  flexibility provides for manual compile-time load balancing across a
 Line 32 
 The exchange parameters are declared in
  and assigned in
  \filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}. The
  validity of the cube topology depends on the \file{SIZE.h} file as
- detailed below.  Both files are generated by Matlab scripts in
+ detailed below.  The default files provided in the release configure a
+ cubed sphere topology of six tiles, one per subdomain, each with
+$\times$32 grid points, all running on a single processor.  Both
+ files are generated by Matlab scripts in
  \file{utils/exch2/matlab-topology-generator}; see Section
  \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
- for details on creating alternate topologies.  The default files
+ for details on creating alternate topologies.  Pregenerated examples
- provided in the release configure a cubed sphere topology of six
+ of these files with alternate topologies are provided under
- tiles, one per subdomain, each with 32$\times$32 grid points, all
- running on a single processor.  Pregenerated examples of these files
- with alternate topologies are provided under
  \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
  file for single-processor execution.
 Line 51 
 met: \\
  $\bullet$ The exch2 package is included when \file{genmake2} is run.
    The easiest way to do this is to add the line \code{exch2} to the
    \file{profile.conf} file -- see Section
-   \ref{sect:buildingCode}\sectiontitle{Building the code} for general
+   \ref{sect:buildingCode} \sectiontitle{Building the code} for general
    details. \\
  $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
 Line 65 
 $\bullet$ An example of \file{W2\_EXCH2\
    configurations other than the one you intend to modify.\\
  $\bullet$ Files containing grid parameters, named
-   \file{tile???.mitgrid} where \file{???} is \file{001} through
+   \file{tile00$n$.mitgrid} where $n$=\code{(1:6)} (one per subdomain),
-   \file{006} (one per subdomain), must be in the working directory
+   must be in the working directory when the MITgcm executable is run.
-   when the MITgcm executable is run.  These files are provided in the
+   These files are provided in the example experiments for cubed sphere
-   example experiments for cubed sphere configurations with
+   configurations with 32$\times$32 cube sides and are non-trivial to
-$\times$32 cube sides and are non-trivial to generate -- please
+   generate -- please contact MITgcm support if you want to generate
-   contact MITgcm support if you want to generate files for other
+   files for other configurations. \\
-   configurations. \\
+ $\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} must
- $\bullet$ As always when compiling MITgcm, the file \file{SIZE.h}
+   be placed where \file{genmake2} will find it.  In particular for
-   must be placed where \file{genmake2} will find it.  In particular
+   exch2, the domain decomposition specified in \file{SIZE.h} must
-   for the exch2, the domain decomposition specified in \file{SIZE.h}
+   correspond with the particular configuration's topology specified in
-   must correspond with the particular configuration's topology
+   \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain
-   specified in \file{W2\_EXCH2\_TOPOLOGY.h} and
+   decomposition issues particular to exch2 are addressed in Section
-   \file{w2\_e2setup.F}.  Domain decomposition issues particular to
+   \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
-   exch2 are addressed in Section \ref{sec:topogen} \sectiontitle{Generating
+   and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and MPI}; a more
-   Topology Files for exch2}; a more general background on the subject
+   general background on the subject relevant to MITgcm is presented in
-   relvant to MITgcm is presented in Section
+   Section \ref{sect:specifying_a_decomposition}
-   \ref{sect:specifying_a_decomposition}\sectiontitle{Specifying a
+   \sectiontitle{Specifying a decomposition}.\\
-   decomposition}.\\
  As of the time of writing the following examples use exch2 and may be
  used for guidance:
-Line 104 
 verification/hs94.cs-32x32x5
+Line 103 
 verification/hs94.cs-32x32x5
  Alternate cubed sphere topologies may be created using the Matlab
  scripts in \file{utils/exch2/matlab-topology-generator}. Running the
- m-file \file{driver.m} from the Matlab prompt (there are no parameters
+ m-file
- to pass) generates exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h}
+ \filelink{driver.m}{utils-exch2-matlab-topology-generator_driver.m}
- and \file{w2\_e2setup.F} in the working directory and displays a
+ from the Matlab prompt (there are no parameters to pass) generates
- figure of the topology via Matlab.  The other m-files in the directory
+ exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and
- are subroutines of \file{driver.m} and should not be run except for
+ \file{w2\_e2setup.F} in the working directory and displays a figure of
- development purposes. \\
+ the topology via Matlab.  The other m-files in the directory are
+ subroutines of \file{driver.m} and should not be run ``bare'' except
+ for development purposes. \\
  The parameters that determine the dimensions and topology of the
  generated configuration are \code{nr}, \code{nb}, \code{ng},
- \code{tnx} and \code{tny}, and all are assigned early in the script.
+ \code{tnx} and \code{tny}, and all are assigned early in the script. \\
- The first three determine the size of the subdomains (cube faces) and
+ The first three determine the size of the subdomains and
  hence the size of the overall domain.  Each one determines the number
  of grid points, and therefore the resolution, along the subdomain
- sides in a ``great circle'' around each axis of the cube.  At the time
+ sides in a ``great circle'' around an axis of the cube.  At the time
  of this writing MITgcm requires these three parameters to be equal,
- but they provide for future releases of MITgcm to accomodate different
+ but they provide for future releases  to accomodate different
  resolutions around the axes to allow (for example) greater resolution
  around the equator.\\
-Line 128 
 The parameters \code{tnx} and \code{tny}
+Line 129 
 The parameters \code{tnx} and \code{tny}
  the tiles into which the subdomains are decomposed, and must evenly
  divide the integer assigned to \code{nr}, \code{nb} and \code{ng}.
  The result is a rectangular tiling of the subdomain.  Figure
- \ref{fig:24tile} shows one possible topology for a twenty-four tile
+ \ref{fig:24tile} shows one possible topology for a twentyfour-tile
  cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\
  \begin{figure}
-Line 137 
 cube, and figure \ref{fig:12tile} shows
+Line 138 
 cube, and figure \ref{fig:12tile} shows
    \includegraphics{part6/s24t_16x16.ps}
   }
  \end{center}
- \caption{Plot of cubed sphere topology with a 32$\times$32 grid and
- twenty-four tiles (\code{tnx=16, tny=16})
+ \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+ divided into six 32$\times$32 subdomains, each of which is divided into four tiles
+ (\code{tnx=16, tny=16}) for a total of twentyfour tiles.
  } \label{fig:24tile}
  \end{figure}
-Line 148 
 twenty-four tiles (\code{tnx=16, tny=16}
+Line 151 
 twenty-four tiles (\code{tnx=16, tny=16}
    \includegraphics{part6/s12t_16x32.ps}
   }
  \end{center}
- \caption{Plot of cubed sphere topology with a 32$\times$32 grid and
+ \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
- twelve tiles (\code{tnx=16, tny=32})
+ divided into six 32$\times$32 subdomains of two tiles each
+  (\code{tnx=16, tny=32}).
  } \label{fig:12tile}
  \end{figure}
+ \begin{figure}
+ \begin{center}
+  \resizebox{4in}{!}{
+   \includegraphics{part6/s6t_32x32.ps}
+  }
+ \end{center}
+ \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+ divided into six 32$\times$32 subdomains with one tile each
+ (\code{tnx=32, tny=32}).  This is the default configuration.
+   }
+ \label{fig:6tile}
+ \end{figure}
  Tiles can be selected from the topology to be omitted from being
- allocated memory and processors.  This kind otuning is useful in
+ allocated memory and processors.  This tuning is useful in ocean
- ocean modeling for omitting tiles that fall entirely on land.  The
+ modeling for omitting tiles that fall entirely on land.  The tiles
- tiles omitted are specified in the file \file{blanklist.txt} by
+ omitted are specified in the file
- their tile number in the topology, separated by a newline. \\
+ \filelink{blanklist.txt}{utils-exch2-matlab-topology-generator_blanklist.txt}
+ by their tile number in the topology, separated by a newline. \\
+ \subsection{exch2, SIZE.h, and multiprocessing}
+ \label{sec:exch2mpi}
+ Once the topology configuration files are created, the Fortran
+ \code{PARAMETER}s in \file{SIZE.h} must be configured to match.
+ Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying
+ a decomposition} provides a general description of domain
+ decomposition within MITgcm and its relation to \file{SIZE.h}. The
+ current section specifies certain constraints the exch2 package
+ imposes as well as describes how to enable parallel execution with
+ MPI. \\
+ As in the general case, the parameters \varlink{sNx}{sNx} and
+ \varlink{sNy}{sNy} define the size of the individual tiles, and so
+ must be assigned the same respective values as \code{tnx} and
+ \code{tny} in \file{driver.m}.\\
+ The halo width parameters \varlink{OLx}{OLx} and \varlink{OLy}{OLy}
+ have no special bearing on exch2 and may be assigned as in the general
+ case. The same holds for \varlink{Nr}{Nr}, the number of vertical
+ levels in the model.\\
+ The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},
+ \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of
+ tiles and how they are distributed on processors.  When using exch2,
+ the tiles are stored in single dimension, and so
+ \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as
+ configured by exch2 cannot be split up accross processors without
+ regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\
+ The number of tiles MITgcm allocates and how they are distributed
+ between processors depends on \varlink{nPx}{nPx} and
+ \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per
+ processor and \varlink{nPx}{nPx} the number of processors.  The total
+ number of tiles in the topology minus those listed in
+ \file{blanklist.txt} must equal \code{nSx*nPx}. \\
+ The following is an example of \file{SIZE.h} for the twelve-tile
+ configuration illustrated in figure \ref{fig:12tile} running on
+ one processor: \\
+ \begin{verbatim}
+       PARAMETER (
+      &           sNx =  16,
+      &           sNy =  32,
+      &           OLx =   2,
+      &           OLy =   2,
+      &           nSx =  12,
+      &           nSy =   1,
+      &           nPx =   1,
+      &           nPy =   1,
+      &           Nx  = sNx*nSx*nPx,
+      &           Ny  = sNy*nSy*nPy,
+      &           Nr  =   5)
+ \end{verbatim}
+ The following is an example for the twentyfour-tile topology in figure
+ \ref{fig:24tile} running on six processors:
+ \begin{verbatim}
+       PARAMETER (
+      &           sNx =  16,
+      &           sNy =  16,
+      &           OLx =   2,
+      &           OLy =   2,
+      &           nSx =   4,
+      &           nSy =   1,
+      &           nPx =   6,
+      &           nPy =   1,
+      &           Nx  = sNx*nSx*nPx,
+      &           Ny  = sNy*nSy*nPy,
+      &           Nr  =   5)
+ \end{verbatim}
  \subsection{Key Variables}
-Line 169 
 their tile number in the topology, separ
+Line 264 
 their tile number in the topology, separ
  The descriptions of the variables are divided up into scalars,
  one-dimensional arrays indexed to the tile number, and two and three
  dimensional arrays indexed to tile number and neighboring tile.  This
- division actually reflects the functionality of these variables: the
+ division reflects the functionality of these variables: The
  scalars are common to every part of the topology, the tile-indexed
- arrays to individual tiles, and the arrays indexed to tile and
+ arrays to individual tiles, and the arrays indexed by tile and
- neighbor to relationships between tiles and their neighbors.
+ neighbor to relationships between tiles and their neighbors. \\
  \subsubsection{Scalars}
  The number of tiles in a particular topology is set with the parameter
- \texttt{NTILES}, and the maximum number of neighbors of any tiles by
+ \code{NTILES}, and the maximum number of neighbors of any tiles by
- \texttt{MAX\_NEIGHBOURS}.  These parameters are used for defining the
+ \code{MAX\_NEIGHBOURS}.  These parameters are used for defining the
  size of the various one and two dimensional arrays that store tile
- parameters indexed to the tile number.\\
+ parameters indexed to the tile number and are assigned in the files
+ generated by \file{driver.m}.\\
  The scalar parameters \varlink{exch2\_domain\_nxt}{exch2_domain_nxt}
  and \varlink{exch2\_domain\_nyt}{exch2_domain_nyt} express the number
- of tiles in the x and y global indices.  For example, the default
+ of tiles in the $x$ and $y$ global indices.  For example, the default
- setup of six tiles has \texttt{exch2\_domain\_nxt=6} and
+ setup of six tiles (Fig. \ref{fig:6tile}) has \code{exch2\_domain\_nxt=6} and
- \texttt{exch2\_domain\_nyt=1}.  A topology of twenty-four square (in
+ \code{exch2\_domain\_nyt=1}.  A topology of twenty-four square tiles,
- gridpoints) tiles, four (2x2) per subdomain, will have
+ four per subdomain (as in figure \ref{fig:24tile}), will have
- \texttt{exch2\_domain\_nxt=12} and \texttt{exch2\_domain\_nyt=2}.
+ \code{exch2\_domain\_nxt=12} and \code{exch2\_domain\_nyt=2}.  Note
- Note that these parameters express the tile layout to allow global
+ that these parameters express the tile layout to allow global data
- data files that are tile-layout-neutral and have no bearing on the
+ files that are tile-layout-neutral and have no bearing on the internal
- internal storage of the arrays.  The tiles are internally stored in a
+ storage of the arrays.  The tiles are internally stored in a range
- range from \texttt{1,bi} (in the x axis) and y-axis variable
+ from \code{(1:\varlink{bi}{bi})} the $x$ axis, and $y$ axis variable
- \texttt{bj} is generally ignored within the package.
+ \varlink{bj}{bj} is generally ignored within the package. \\
  \subsubsection{Arrays Indexed to Tile Number}
- The following arrays are of size \texttt{NTILES}, are indexed to the
+ The following arrays are of length \code{NTILES}, are indexed to the
- tile number, and the indices are omitted in their descriptions.
+ tile number, and the indices are omitted in their descriptions. \\
  The arrays \varlink{exch2\_tnx}{exch2_tnx} and
- \varlink{exch2\_tny}{exch2_tny} express the x and y dimensions of each
+ \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of
- tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
+ each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
- \texttt{exch2\_tny=sNy}, as assigned in \texttt{SIZE.h}.  Future
+ \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in
- releases of MITgcm are to allow varying tile sizes.
+ section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
+ multiprocessing}.  Future releases of MITgcm are to allow varying tile
+ sizes. \\
  The location of the tiles' Cartesian origin within a subdomain are
  determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
  \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to
- relate the location of the edges of the tiles to each other.  As an
+ relate the location of the edges of different tiles to each other.  As
- example, in the default six-tile topology (the degenerate case) each
+ an example, in the default six-tile topology (Fig. \ref{fig:6tile})
- index in these arrays are set to 0.  The twenty-four, 32x32 cube face
+ each index in these arrays is set to \code{0} since a tile occupies
- case discussed above will have values of 0 or 16, depending on the
+ its entire subdomain.  The twentyfour-tile case discussed above will
- quadrant the tile falls within the subdomain.  The array
+ have values of \code{0} or \code{16}, depending on the quadrant the
- \varlink{exch2\_myFace}{exch2_myFace} contains the number of the
+ tile falls within the subdomain.  The elements of the arrays
- cubeface/subdomain of each tile, numbered 1-6 in the case of the
+ \varlink{exch2\_txglobalo}{exch2_txglobalo} and
- standard cube topology.
- The arrays \varlink{exch2\_txglobalo}{exch2_txglobalo} and
  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to
  \varlink{exch2\_tbasex}{exch2_tbasex} and
  \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the
- global address space, similar to that used by global files.
+ global address space, similar to that used by global files. \\
+ The array \varlink{exch2\_myFace}{exch2_myFace} contains the number of
+ the subdomain of each tile, in a range \code{(1:6)} in the case of the
+ standard cube topology and indicated by \textbf{\textsf{f}}$n$ in
+ figures \ref{fig:12tile} and
+ \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
+ contains a count of how many neighboring tiles each tile has, and is
+ used for setting bounds for looping over neighboring tiles.
+ \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
+ tile, and is used in interprocess communication.  \\
  The arrays \varlink{exch2\_isWedge}{exch2_isWedge},
  \varlink{exch2\_isEedge}{exch2_isEedge},
  \varlink{exch2\_isSedge}{exch2_isSedge}, and
- \varlink{exch2\_isNedge}{exch2_isNedge} are set to 1 if the indexed
+ \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
- tile lies on the edge of a subdomain, 0 if not.  The values are used
+ indexed tile lies on the edge of a subdomain, \code{0} if not.  The
- within the topology generator to determine the orientation of
+ values are used within the topology generator to determine the
- neighboring tiles and to indicate whether a tile lies on the corner of
+ orientation of neighboring tiles, and to indicate whether a tile lies
- a subdomain.  The latter case indicates special exchange and numerical
+ on the corner of a subdomain.  The latter case requires special
- handling for the singularities at the eight corners of the cube.
+ exchange and numerical handling for the singularities at the eight
- \varlink{exch2\_nNeighbours}{exch2_nNeighbours} contains a count of
+ corners of the cube. \\
- how many neighboring tiles each tile has, and is used for setting
- bounds for looping over neighboring tiles.
- \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
- tile, and is used in interprocess communication.
- \subsubsection{Arrays Indexed to Tile Number and Neighbor}
- The following arrays are all of size \texttt{MAX\_NEIGHBOURS} $\times$
+ \subsubsection{Arrays Indexed to Tile Number and Neighbor}
- \texttt{NTILES} and describe the orientations between the the tiles.
- The array \texttt{exch2\_neighbourId(a,T)} holds the tile number for
+ The following arrays are all of size
- each of the $n$ neighboring tiles.  The neighbor tiles are indexed
+ \code{MAX\_NEIGHBOURS}$\times$\code{NTILES} and describe the
- \texttt{(1,MAX\_NEIGHBOURS} in the order right to left on the north
+ orientations between the the tiles. \\
- then south edges, and then top to bottom on the east and west edges.
- Maybe throw in a fig here, eh?
+ The array \code{exch2\_neighbourId(a,T)} holds the tile number
+ \code{Tn} for each of the tile number \code{T}'s neighboring tiles
- The \texttt{exch2\_opposingSend\_record(a,T)} array holds the index c
+ \code{a}.  The neighbor tiles are indexed \code{(1:MAX\_NEIGHBOURS)}
- in \texttt{exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T.
+ in the order right to left on the north then south edges, and then top
- In other words,
+ to bottom on the east and west edges.  Maybe throw in a fig here, eh?
+ \\
+ \sloppy
+ The \code{exch2\_opposingSend\_record(a,T)} array holds the index
+ \code{b} in \texttt{exch2\_neighbourId(b,Tn)} that holds the tile
+ number \code{T}.  In other words,
  \begin{verbatim}
     exch2_neighbourId( exch2_opposingSend_record(a,T),
                        exch2_neighbourId(a,T) ) = T
  \end{verbatim}
- and this provides a back-reference from the neighbor tiles.
+ This provides a back-reference from the neighbor tiles. \\
+ The arrays \varlink{exch2\_pi}{exch2_pi} and
+ \varlink{exch2\_pj}{exch2_pj} specify the transformations of variables
+ in exchanges between the neighboring tiles.  These transformations are
+ necessary in exchanges between subdomains because a physical vector
+ component in one direction may map to one in a different direction in
+ an adjacent subdomain, and may be have its indexing reversed. This
+ swapping arises from the ``folding'' of two-dimensional arrays into a
+ three-dimensional cube.
+ The dimensions of \code{exch2\_pi(t,N,T)} and \code{exch2\_pj(t,N,T)}
+ are the neighbor ID \code{N} and the tile number \code{T} as explained
+ above, plus a vector of length 2 containing transformation factors
+ \code{t}.  The first element of the transformation vector indicates
+ the factor \code{t} by which variables representing the same
+ \emph{physical} vector component of a tile \code{T} will be multiplied
+ in exchanges with neighbor \code{N}, and the second element indicates
+ the transform to the physical vector in the other direction.  To
+ clarify (hopefully), \code{exch2\_pi(1,N,T)} holds the transform of
+ the $i$ component of a vector variable in tile \code{T} to the $i$
+ component of tile \code{T}'s neighbor \code{N}, and
+ \code{exch2\_pi(2,N,T)} holds the transform of \code{T}'s $i$
+ components to the neighbor \code{N}'s $j$ component. \\
+ Under the current cube topology, one of the two elements of
+ \code{exch2\_pi} or \code{exch2\_pj} for a given tile \code{T} and
+ neighbor \code{N} will be \code{0}, reflecting the fact that the two
+ vector components are orthogonal.  The other element will be \code{1}
+ or \code{-1}, depending on whether the components are indexed in the
+ same or opposite directions.  For example, the transform vector of the
+ arrays for all tile neighbors on the same subdomain will be
+ \code{(1,0)}, since all tiles on the same subdomain are oriented
+ identically.  A vector direction that corresponds to the orthogonal
+ dimension with the same index direction in a particular tile-neighbor
+ orientation will have \code{(0,1)}, whereas those in the opposite
+ index direction will have \code{(0,-1)}. \\
- The arrays \varlink{exch2\_pi}{exch2_pi},
+ \varlink{exch2\_oi}{exch2_oi},
- \varlink{exch2\_pj}{exch2_pj}, \varlink{exch2\_oi}{exch2_oi},
  \varlink{exch2\_oj}{exch2_oj}, \varlink{exch2\_oi\_f}{exch2_oi_f}, and
- \varlink{exch2\_oj\_f}{exch2_oj_f} specify the transformations in
+ \varlink{exch2\_oj\_f}{exch2_oj_f}
- exchanges between the neighboring tiles.  The dimensions of
- \texttt{exch2\_pi(t,N,T)} and \texttt{exch2\_pj(t,N,T)} are the
- neighbor ID \textit{N} and the tile number \textit{T} as explained
- above, plus the transformation vector {\em t }, of length two.  The
- first element of the transformation vector indicates the factor by
- which variables representing the same vector component of a tile will
- be multiplied, and the second element indicates the transform to the
- variable in the other direction.  As an example,
- \texttt{exch2\_pi(1,N,T)} holds the transform of the i-component of a
- vector variable in tile \texttt{T} to the i-component of tile
- \texttt{T}'s neighbor \texttt{N}, and \texttt{exch2\_pi(2,N,T)} hold
- the component of neighbor \texttt{N}'s j-component.
- Under the current cube topology, one of the two elements of
- \texttt{exch2\_pi} or \texttt{exch2\_pj} for a given tile \texttt{T}
- and neighbor \texttt{N} will be 0, reflecting the fact that the vector
- components are orthogonal.  The other element will be 1 or -1,
+ This needs some diagrams. \\
- depending on whether the components are indexed in the same or
- opposite directions.  For example, the transform dimension of the
- arrays for all tile neighbors on the same subdomain will be [1,0],
- since all tiles on the same subdomain are oriented identically.
- Vectors that correspond to the orthogonal dimension with the same
- index direction will have [0,1], whereas those in the opposite index
- direction will have [0,-1].
  {\footnotesize

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+Added in v.1.13

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