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\documentclass{article}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage[english] {babel}
\usepackage{exscale}
\usepackage[final]{graphicx}
\usepackage[backref,draft=false]{hyperref}
%\usepackage{concrete}
\usepackage{mathpple}
%\usepackage{pslatex}
\newcommand{\todo}[1]{\rule{1em}{1ex}~{\small [{#1}]}}
\sloppypar
\begin{document}
\title{Carpet under the hood}
\author{Erik Schnetter \textless schnetter@uni-tuebingen.de\textgreater}
\date{May 3, 2003}
\maketitle
\begin{abstract}
This document describes the internal workings of the Carpet
arrangement. Its intended readership are people who extend Carpet,
or who use Carpet more thant the average user. This document is
supposed to be read in conjuction with and guiding through the
source code.
\end{abstract}
\tableofcontents
\section{Overview}
The Carpet driver, which lives in the Carpet arrangement, is
divided into several parts. The thorn \texttt{Carpet} is the main
driver piece; it provides all the routines and structures that
Cactus expects from it. The thorn \texttt{CarpetLib} is the
workhorse that does all the bookkeeping and data shuffling. Those
two alone form a valid Cactus driver; the other thorns provide
additional functionality. The thorns \texttt{CarpetInterp},
\texttt{CarpetReduce}, and \texttt{CarpetSlab} provide the
corresponding interpolation, reduction, and slabbing interfaces.
The thorns \texttt{CarpetIOASCII} and \texttt{CarpetIOFlexIO}
provide I/O methods. Finally, thorn \texttt{CarpetRegrid} provides
a user interface to select where and what to refine. (The actual
refinement is handled in \texttt{CarpetLib}.)
\section{Terminology}
Carpet is called ``Carpet'' because a carpet consists of many
individual patches.
Carpet is a mesh refinement driver. It knows about a hierarchy of
\emph{refinement levels}, where each level is decomposed into a set
of cuboid \emph{grid patches}. For historic reasons it also has a
notion of \emph{multigrid levels}, but those are currently unused.
They might conceivably be reactivated to form multigrid stacks to
solve elliptic equations. The grid patch is the smallest unit of
grid points that Carpet deals with. Carpet parallelises by
assigning sets of grid patches to processors.
A multi-patch run is a run where more than one grid patch (of the
same refinement level) is assigned to a single processor. This is
a situation that can occur even without refinement. This is also a
situation that cannot occur with PUGH, so that most thorns cannot
handle this situation. In multi-patch runs one has to distinguish
between \emph{local mode}, where one has access to a single grid
patch, and \emph{global mode}, where one cannot access individual
grid patches, but can instead perfom global operations such as
synchronisation, interpolation, or reduction. This part of Cactus
is currently (2003-04-30) undergoing changes.
Carpet uses vertex-centered refinement. That is, each coarse grid
point coincides with a fine grid point. To \emph{regrid} means to
select a new set of grid patches for each refinement level. To
\emph{recompose} the grid hierarchy means to move data around.
Regridding is only about bookkeeping, while recomposing is about
data munging.
Each grid patch can be divided in up to four zones: the interior,
the outer boundary, and the ghost zone, and the refinement
boundary. The interior is where the actual compuations go on. The
outer boundary is where the users' outer boundary condition is
applied; from Carpet's point of view, these two are the same. (The
only difference is that Carpet sets \texttt{cctk\_bbox}
correspondingly.) The ghost zones are boundaries to other grid
patches on the same refinement level (that might live on a
different processor). The refinement boundary is the boundary of
the refined region in a level, and it is filled by prolongation
(interpolation) from the next coarser level. Both the ghost zones
and the prolongation boundary are filled by \emph{synchronising}.
Grid patches that are on the same refinement level never overlap
except with their ghost zones. Conversly, all ghost zones must
overlap with a non-ghost zone of another grid patch of the same
level. All refinement boundaries must overlap with a grid patch on
the next coarser level. (This is also called \emph{proper
nesting}.)
Except for exceptions, Carpet numbers grid point indices and time
levels with integers. It counts always in terms of the finest
grid, so that coarser grids have \emph{strides} that are powers of
the refinement factor. This has the advantage that different
refinement levels can use the same global numbering scheme.
The grid patches are described by a \emph{bounding box}
(abbreviated bbox, see \texttt{CarpetLib/src/bbox.*}.). This is a
triplet of \emph{vectors} (see \texttt{CarpetLib/src/vect.*}),
where each triplet specifies \emph{lower bound}, \emph{upper
bound}, and \emph{stride}, much as is conventional in Fortran.
Triplets are enclosed in round parentheses $(\cdot:\cdot:\cdot)$,
and vectors are enclosed in square brackets $[\cdot,\cdot,\cdots]$.
A typical grid patch might have a bounding box which is denoted by
$([0,0,0]:[20,20,20]:[2,2,2])$. This is to be read as
$(\textrm{lower}:\textrm{upper}:\textrm{stride})$, meaning that the
grid patch has one corner grid point at $[0,0,0]$, the diagonally
opposite corner grid point at $[20,20,20]$, and the grid points are
spaced two ``fine grid spacings'' apart. This grid patch contains
$11 \times 11 \times 11$ grid points. Empty bboxes have an upper
bound that is strictly lower than the lower bound. The files
\texttt{CarpetLib/src/vect.*} contains many useful routines to deal
with short vectors, and the files \texttt{CarpetLib/src/bbox.*}
contain routines deal with an algebra of bboxes. The files
\texttt{CarpetLib/src/bboxset.*} contain routines that handle sets
of bboxes.
\section{The driver}
The driver consists of the two thorns \texttt{Carpet} and
\texttt{CarpetLib}. \texttt{Carpet} is the front end to
Cactus, while \texttt{CarpetLib} is the back end to the
machine. \texttt{Carpet} specifies the grid shape, decides when to
allocate and deallocate storage, cycles through thes schedule bins,
and passes all internal information in the \texttt{cGH} structure
to the thorns.
\subsection{Specifying the grid extent}
\texttt{Carpet} defines the usual parameters necessary to specify
the extent of the grid. Everything that has to do with coordinates
and symmetries is handled elsewhere, and the driver does not know
about that.
The \texttt{global\_*} parameters specify the global extent of the
coarsest grid. Not all of this grid needs to be covered by grid
patches. It is conceivable to have an L-shaped simulation domain
without any refinement. This situation can be described to Carpet
by specifying a global shape that is the convex hull of the domain,
and then using two cuboid grid patchs to fill in the shape of
the~L.
The \texttt{ghost\_*} parameters specify the number of ghost zones.
The \texttt{periodic*} parameters are unused; they are only there
because some thorns look at these parameters. Carpet itself does
not supply periodic boundary conditions; they have to be handled by
another thorn. The size of the prolongation boundary is the same
as the number of ghost zones.
The parameter \texttt{max\_refinement\_levels} specifies the
maximum number of levels that can be present in a run, including
the base level. This parameter, together with
\texttt{refinement\_factor}, define the grid point numbering
scheme, which (see above) depends on the finest possible grid.
However, none of the finer levels will be activated automatically.
The \texttt{multigrid\_*} parameters are unused.
The parameter \texttt{base\_extents} specifies the shapes of the
grid patches that are present on the coarsest grid. This can be
used to set up e.g.\ an L-shaped domain. The parameter
\texttt{base\_outerbounds} specifies which of the grid patches'
boundaries are to be treated as outer boundaries, i.e.\ for which
of those \texttt{cctk\_bbox} should be set to 1.
Carpet currently ignores \texttt{enable\_all\_storage} and always
enables all storage. This is because it is not yet clear how
individual grid function can be allocated and deallocated while
still keeping enough data for the boundary prolongation.
Checksumming and poisoning are means to find thorns that alter grid
variables that should not be altered, or that fail to fill in grid
variables that they should fill in.
None of the above specifies anything about refined grids. Refined
grid are created and destroyed at run time, possibly guided by the
thorn \texttt{CarpetRegrid} which provides a nice user interface.
\subsection{The timeline}
It is \texttt{Carpet}'s task to walk through the schedule bins and
call all user routines. Only some fairly fundamental
initialisation happens in the flesh before Carpet takes control.
The overall picture of what happens when is:
\begin{enumerate}
\item
Startup (see file \texttt{Carpet/src/CarpetStartup.cc}). This is
the only scheduled routine; everything else happens by overloading
and registering. This routine does nothing but registering and
overloading.
\item
SetupGH (see file \texttt{Carpet/src/SetupGH.cc}). This routine
does the bulk of initialising Carpet. It sets up the internal
structures for all grid variables.
\item
Initialise (see file \texttt{Carpet/src/Initialise.cc}). This
routine walks the initialisation part of the scheduling bins.
\item
Evolve (see file \texttt{Carpet/src/Evolve.cc}). This routine
walks the evolution part of the scheduling bins. It also contains
the main evolution loop.
\item
Shutdown (see file \texttt{Carpet/src/Shutdown.cc}). This routine
walks the shutdown part of the scheduling bins. Normally, nothing
interesting happens here.
\end{enumerate}
These stages are explained in the following sections.
\subsubsection{Initialisation}
(See file \texttt{Carpet/src/Initialise.cc}.) In this stage Carpet
initialises the simulation. This includes setting up the grids,
calling routines to register symmetries and boundary conditions, as
well as calculating the actual initial data on several refinement
levels.
There are three parameters influencing initial data generation, and
it does not make sense to set more than one to "yes":
\begin{verbatim}
MoL::initial_data_is_crap
Carpet::init_each_timelevel
Carpet::init_3_timelevels
\end{verbatim}
That is, you have four methods, and the default (all no) gives you
wrong data on the past timelevels and hence wrong data on the
interpolated refinement boundaries when you use second order time
interpolation. For first order time interpolation, all four methods
are identical.
With all three parameters set to "no"
Carpet traverses the scheduling bins in the following order:
\begin{enumerate}
\itemsep 0pt
\item
Set \texttt{cctk\_iteration} to zero
\item
Set \texttt{cctk\_time} to the initial time
\item
PARAMCHECK
\item
Loop over refinement levels, starting from coarsest:
%\begin{enumerate}
\item \quad
BASEGRID
\label{startinitsubloop}
\item \quad
INITIAL
\item \quad
POSTINITIAL
\item \quad
POSTSTEP
\label{almostendinitsubloop}
\item \quad
Regrid (possibly creating new levels)
\label{endinitsubloop}
%\end{enumerate}
\item
End loop over refinement levels
\item
Restrict from finer to coarser grids
%\item
% If desired, perform Scott Hawley's initialisation scheme for three
% timelevels
\item
Loop over refinement levels, starting from coarsest:
%\begin{enumerate}
\item \quad
RECOVER\_VARIABLES
\item \quad
CPINITIAL
\item \quad
ANALYSIS
\item \quad
OutputGH
%\end{enumerate}
\item
End loop over refinement levels
\label{endinitloop}
\end{enumerate}
In the beginning, only the coarsest level exists. The first loop
starts by initialising this level. At the end of this loop, more
levels are created if desired. This makes it possible to make this
decision depend on an automatic refinement criterion.
\texttt{ MoL::initial\_data\_is\_crap} performs all steps as indicated.
%, except 12
After \ref{endinitloop}, when the evolution starts, \texttt{MoL} copies the
current
timelevels to the past timelevels.
\texttt{Carpet::init\_each\_timelevel} loops the steps \ref{startinitsubloop}
to
\ref{almostendinitsubloop} over all
timelevels, setting \texttt{cctk\_time} differently each time.
Finally, \texttt{Carpet::init\_3\_timelevels} performs all steps in order, but
evolves each level forward one and backwards two steps, creating two
past time levels.
The parameter that specifies the number of refinement levels is not a
Carpet parameter, but a CarpetRegrid parameter. CarpetRegrid
determines item \ref{endinitsubloop}, i.e., whether to create a new, finer
level when
the coarser levels have been initialised. CarpetRegrid has a host of
other parameters, and it can decide item \ref{endinitsubloop} also by a
different means,
e.g. ---in principle--- through the local truncation error.
\subsubsection{Evolution}
(See file \texttt{Carpet/src/Evolve.cc}.) In this stage Carpet
performs the main time evolution loop. This is further complicated
by the fact that finer grids need to take more and smaller time
steps than coarser grids. In Carpet's time step counting scheme,
which is based on the finest grid time steps, this means that the
coarser grids are skipped in the remaining time steps. Thus the
elegant recursive scheme is flattened out. The scheduling bins in
the main time evolution loop are traversed in the following order:
\begin{enumerate}
\itemsep 0pt
\item
Advance \texttt{cctk\_iteration}
\item
Loop over refinement levels, starting from coarsest:
\item \quad
If the current level needs to be treated at this iteration:
\item \quad \quad
Calculate current \texttt{cctk\_time}
\item \quad \quad
Cycle time levels
\item \quad \quad
PRESTEP
\item \quad \quad
EVOL
\item \quad \quad
POSTSTEP
\item \quad \quad
Regrid
\item
End loop over refinement levels
\item
Restrict from finer to coarser grids
\item
Loop over refinement levels, starting from coarsest:
\item \quad
If the current level needs to be treated at this iteration:
\item \quad \quad
CHECKPOINT
\item \quad \quad
ANALYSIS
\item \quad \quad
OutputGH
\item
End loop over refinement levels
\end{enumerate}
The condition whether a refinement level needs to be treated at the
current iteration is different for the two loops. In the first
loop, the coarse grids need to be advanced before the finer grids,
so the condition is $iter \,\mathrm{mod}\, stride = 1$. Here
$iter$ is the current iteration, and $stride$ the stride of the
current refinement level, i.e.\ the factor by which the finest grid
is finer than the current grid. In the second loop above, the
coarse grids need to be treated after the finer grids, so that the
condition reads $iter \,\mathrm{mod}\, stride = stride$.
\subsection{Calling scheduled routines}
(See file \texttt{Carpet/src/CallFunction.cc}.) The process by
which the scheduling bins are traversed is different from the
process which actually calls the routines within the scheduling
bins. The former has to do with mesh refinement, making sure that
the coarse and fine grids are evolved in the right order. The
latter has to do with treating multiple patches, i.e.\ with local
mode and global mode operations, as mentioned above.
In the function \texttt{CallFunction}, all the arguments that are
passed to the scheduled routines have to be set up. Additionally,
the \texttt{cGH} structure has to be filled in. Some fields in the
\texttt{cGH} structure are always kept up-to-date during the
refinement level loops, such as the time step size and the grid
spacing. The file \texttt{Carpet/src/helper.cc} contains helper
routines that allow easy looping over refinement levels and over
grid patches. (Grid patches are also called \emph{compoments} in
Carpet. The expression component seems to be confusing, so that I
switched to using \emph{patch} instead. Some source code still
reflects the old conventsion.)
The function \texttt{CallFunction} first distinguishes between
global mode functions and local mode functions.
\begin{description}
\item[Global mode functions]
are called once (on each processor). They are passed all the
global data, such as \texttt{cctk\_gsh} and
\texttt{cctk\_delta\_space}, but none of the local data, such as
\texttt{cctk\_lsh} or \texttt{cctk\_bbox}. Grid functions are not
accessible, and they are passed as null pointers. However, grid
scalars and grid arrays are accessible. There is an untested
gateway to directly call local mode functions from global mode
functions.
\item[Local mode functions]
might be called several times (on each processor), once for each
grid patch that is assigned to this processor. They receive the
global data as well as data for a single grid patch. It is illegal
to perform global operations, such as synchronisation,
interpolation, or reduction, in local mode.
\end{description}
The distinction between global and local mode is only important for
multi-patch runs. For single-patch runs, the distinction does not
exist.
Multi-patch runs are only necessary when there are more grid
patches on a refinement level than there are processors. This is
normally not the case for fixed mesh refinement. Things are
different for adaptive mesh refinement, which can create many
refined regions.
\subsection{Grid arrays and grid scalars}
Grid scalars are implemented as zero-dimensional grid arrays with
\texttt{DISTRIB=CONSTANT}.
Grid arrays are implemented as grid functions, where each grid
array group has their own refinement hierarchy that consists of a
single level only and is never changed at run time. Grid arrays
with less than 3 dimension are extended to have an extent of 1 (and
no ghost zones) in the remaining dimensions, so that all quantities
in Carpet have 3 dimensions\footnote{This is set by a compile-time
constant and could be changed to allow for grid functions and
arrays with more than 3 dimensions.}. \texttt{DISTRIB=CONSTANT} grid arrays
are implemented by internally enlarging the grid array in the $z$
direction, and then distributing this array onto the processors.
\subsection{Flesh interfaces}
The flesh has many interfaces that need to be filled in by a
driver. These are in particular all the routines that are
overloaded in the SetupGH stage. Those overloaded routines as well
as other helper function are implemented in the following files:
\begin{description}
\itemsep 0pt
\item[\texttt{Carpet/src/Checksum.cc}]
catching illegal changes to grid variables
\item[\texttt{Carpet/src/Comm.cc}]
synchronisation, prolongation
\item[\texttt{Carpet/src/Cycle.cc}]
time level handling
\item[\texttt{Carpet/src/Poison.cc}]
catching uninitialised grid variables
\item[\texttt{Carpet/src/Restrict.cc}]
restriction from finer to coarser grids
\item[\texttt{Carpet/src/Storage.cc}]
enabling and disabling storage
\item[\texttt{Carpet/src/helpers.cc}]
small low-level helper routines
\item[\texttt{Carpet/src/variables.cc}]
the global variables that keep Carpet's current state (this is used
instead of a GH extension --- should probably be changed some time)
\end{description}
Most of these files are fairly self-contained, and they mostly
marshal the actual work to \texttt{CarpetLib}.
\subsection{Interfaces to other thorns}
Some other thorns, mostly from the Carpet arrangement, do need to
access internal data of Carpet. Carpet keeps its internal state in
global variables which are declared in
\texttt{Carpet/src/carpet\_public.hh} and defined in
\texttt{Carpet/src/variables.cc}. Entities that can be accessed
from C are declared in \texttt{Carpet/src/carpet\_public.h}; some
of these would be quite useful if they were provided by the flesh.
\subsection{Missing parts}
\texttt{Carpet} does not handle staggered grids. \texttt{Carpet}
does not provide cell-centered refinement. \texttt{Carpet} always
enables all storage. \texttt{Carpet} does not run efficiently in
parallel.
\section{The workhorse}
While \texttt{Carpet} provides the necessary interfaces to the
flesh, the grunt work is actually done by \texttt{CarpetLib}. This
thorn grew from an earlier mesh refinement of mine (Erik Schnetter)
library that was independent of Cactus. It has in the mean time
been thoroughly changed, and it does not make sense any more to use
it independent of Cactus. \texttt{CarpetLib} contains of three
major parts: a set of generic useful helpers, the grid hierarchy
and data handling, and interpolation operators. Especially the
latter could probably be separated out. While \texttt{CarpetLib}
is written in C++, the interpolators are written in
\textsc{Fortran77}.
\subsection{The helpers}
The helpers correspond closely to Carpet's terminology. A class
\texttt{vect<T,D>} provides small \texttt{D}-dimensional vectors of
the type \texttt{T}, with all the operators that one has learned to
enjoy from Haskell and Fortran 90. A \texttt{vect} corresponds to
a grid point location. The class \texttt{bbox<T,D>} provides
\texttt{D}-dimensional bounding boxes using type \texttt{T} as
indices. A \texttt{bbox} defines the location and shape of a grid
patch. Finally, \texttt{bboxset<T,D>} is a collection of \texttt{bbox}es.
\texttt{bboxsets} are a useful extension of the algebra of bboxes, as it
closes the \texttt{bbox} algebra under the union operation.
The files \texttt{CarpetLib/src/defs.*} defines useful small
helpers and instantiates the STL templates.
\texttt{CarpetLib/src/dist.*} provides some routines around MPI.
Carpet is closely coupled to MPI and does not work without it.
(Instead of inserting switches into Carpet to make it work without
MPI, it would make more sense to use a dummy version of MPI. PETSc
does contain such a dummy version. It is also easily possible to
use a free MPI version such as MPICH and use that to run on a
single processor. However, I cannot see any real need for making
Carpet work without MPI.)
\subsection{The grid hierarchy}
The grid hierarchy is described by a set of classes. Except for
the actual data, all structures and all information is replicated
on all processors.
\begin{description}
\item[\texttt{gh}]
is a grid hierarchy. It describes, for each refinement level, the
location of the grid patches. This \texttt{gh} does not contain
ghost zones or prolongation boundaries. There exists only one
common \texttt{gh} for all grid functions.
\item[\texttt{dh}]
is a data hierarchy. It extends the notion of a \texttt{gh} by
ghost zones and prolongation boundaries. The \texttt{dh} does most
of the bookkeeping work, deciding which grid patches interact with
what other grid patches through synchronisation, prolongation,
restriction, and boundary prolongation. Unexpected situations are
often caught in one of \texttt{dh}'s many self checks. As all grid
functions have the same number of ghost zones, there exists also
only one \texttt{dh} for all grid functions.
\item[\texttt{th}]
is a time hierarchy. It extends the notion of a \texttt{gh} by
multiple time levels. There exists one \texttt{th} per grid
function group. This is a small class that keeps track of the
current time on the different refinement levels. (Note that
different refinement levels usually live at different times.)
\item[\texttt{gf}]
is a grid function of a certain variable type. There is one
instance of \texttt{gf} for every grid function, whether it has
storage or not. Each \texttt{gf} is associated with a \texttt{dh}
and a \texttt{th} and holds the storage for all levels and all
patches. It provides interfaces to access and modify these data,
either directly or through interpolation operators. \texttt{gf}
also handles the data movement during a regridding operation.
\item[\texttt{ggf}]
is an abstract superclass of \texttt{gf} which is independent of
the variable type. This is necessary in C++ to prevent egregious
code duplication due to class templates. Most of the routines in
\texttt{gf} are actually declared in \texttt{ggf}, and they either
are virtual functions themselves, or they call virtual functions
that are declared in \texttt{gf}.
\item[\texttt{data}]
is a container for a grid patch of a certain variable type. This is
a glorified multi-dimensional array that knows how to move between
processors. \texttt{data} is not only used to store the grid
patches that make up a \texttt{gf}, it is also used to move parts
of patches around, e.g.\ for synchronisation or prolongation.
\item[\texttt{gdata}]
is an abstract superclass of \texttt{data} for much the same
reasons as for \texttt{ggf}. All information that is independent
of the variable type is kept in \texttt{gdata}.
\end{description}
\subsection{The interpolators}
There are three kinds of ``interpolators'': for prolongation, for
restricting, and for copying. The latter is only a glorified
hyperslabber that moves parts of grid patches between grid patches.
The interpolators used for restriction and prolongation are
different from those used for the generic interpolation interface
in Cactus. The reason is that interpolation is expensive, and
hence the interpolation operators used for restriction and
prolongation have to be streamlined and optimised. As one knows
the location of the sampling points for the interpolation, one can
calculate the coefficients in advance, saving much time compared to
calling a generic interpolation interface.
\subsubsection{Restriction}
Restriction operators move data from finer to coarser grids. They
are typically called after both the coarse and the fine grid have
been advanced to the same time, and they overwrite parts of the
coarse grid with information from the fine grid, coupling the
coarse grid evolution to the fine grid evolution. In principle,
there could be restriction operators with different orders of
accuracy. Currently only a single restriction operator is
implemented that uses sampling.
The interface of the restriction operator (see file
\texttt{CarpetLib/src/restrict\_3d\_real8.F77}) is
\begin{verbatim}
subroutine restrict_3d_real8
(src, srciext, srcjext, srckext,
dst, dstiext, dstjext, dstkext,
srcbbox, dstbbox, regbbox)
integer srciext, srcjext, srckext
CCTK_REAL8 src(srciext,srcjext,srckext)
integer dstiext, dstjext, dstkext
CCTK_REAL8 dst(dstiext,dstjext,dstkext)
integer srcbbox(3,3), dstbbox(3,3), regbbox(3,3)
\end{verbatim}
This interpolator assumes that space has three dimensions. The
arrays \texttt{src} and \texttt{dst} contain the source (fine) and
destination (coarse) grid patches, stored in Fortran order, as is
customary in Cactus. The arrays \texttt{src} and \texttt{dst} have
the shapes \texttt{(srciext,srcjext,srckext)} and
\texttt{(dstiext,dstjext,dstkext)}, respectively --- this
corresponds to the \texttt{cctk\_lsh} field in the \texttt{cGH}
structure.
The three bboxes describe the location and shape of the two arrays
and of the region that should be prolongated in the global grid
point index system. That is, while the two arrays \texttt{src} and
\texttt{dst} are stored as dense arrays, they correspond to grid
patches which in general have non-unit strides in the global index
system. As prolongation is an operation that is performed between
overlapping grids, the prolongation region is the same for both the
coarse and the fine grids.
A few constraints must hold for these data. For example, the
shapes of the arrays must be the same as the shapes defined by the
bounding boxes; the strides in the bounding boxes must differ by
the refinement factor; the bounding boxes must overlap, and the
region's bounding box must be contained in the arrays bounding
boxes, etc. Checking these constraints makes up about three
quarters of the restriction routine.
The bboxes themselves are here represented as Fortran arrays.
Their meaning is
\begin{description}
\itemsep 0pt
\item[\texttt{bbox(:,1)}]
lower boundary (inclusive)
\item[\texttt{bbox(:,2)}]
upper boundary (inclusive)
\item[\texttt{bbox(:,3)}]
stride
\end{description}
\subsubsection{Prolongation}
There are many prolongation operators implemented. They differ in
the order of their interpolation in space (first and third, or
linear and cubic interpolation) and in time (first and second, or
linear and quadratic). The higher the order of interpolation, the
larger is the stencil, i.e.\ the more ghost zones and time levels
are necessary, and the more expensive the operation becomes.
The prolongation operators live in the files
\texttt{CarpetLib/src/prolongate\_3d\_real8*.F77}, and the file
names indicate their orders: \texttt{$n$tl} stands for $n$ time
levels, and \texttt{o$n$} stands for an order $n$ interpolation in
space (which uses a stencil that is $n+1$ grid points wide).
Apart from taking more than one \texttt{src} array argument when
using more than one time level, the interface to the prolongation
operator is equivalent to that of the restriction operator
described above.
\section{Regridding, how and where and when}
The thorn \texttt{Carpet} provides a routine
\texttt{RegisterRegridRoutine} where one can register a regridding
routine. Such a regridding routine does not have to actually
regrid anything, it only has to return the new desired grid
hierarchy, i.e.\ basically a description of a \texttt{gh}.
Thorn \texttt{CarpetRegrid} provides a user interface to the
regridding routines in Carpet. All it does is take a regridding
specification from the user and translate that into a \texttt{gh}.
As usual, the parts where the computer has to listen to what a
human being intends are the most complicated.
As humans are usually more adept at getting used to computers than
the other way around, it is useful and probably necessary to get
acquainted with how Carpet thinks in order to make it do what is
intended.
Carpet does not deal with real-valued coordinates. Carpet deals
with integer grid point locations only, and it counts grid points
in terms of the finest possible grid (not the finest currently
existing grid). The finest possible grid is defined by the maximum
number of refinement levels set in \texttt{Carpet}. Changing this
parameter will change the meaning of many other values in parameter
files, such as e.g.\ iteration numbers (termination, output). The
only parameter that is specified in terms of the coarsest grid is
the shape of the coarsest grid in the \texttt{global\_*} parameters
of \texttt{Carpet}. I therefore suggest to set
\texttt{max\_refinement\_levels} to some large number (e.g.\ 10),
and then not changing it while experimenting with other parameter
settings.
Carpet also does not know about symmetries. When specifying the
location of a fine grid in terms of grid points, it is the
responsibility of the user to place the fine grid correctly. For
that one has to take ghost zones and symmetry zones into account.
It is also possible to specify the fine grid locations in terms of
real-valued coordinates. In this case, \texttt{CarpetRegrid}
translates these into integer grid points. A good translation is
quite complicated, because it has to take many user expectations
into account, such as the location of the origin, staggering with
respect to the origin, symmetry boundary conditions, the number of
ghost zones etc. The current translation is naive and leads to
unexpected results in many cases. A routine that does most of the
time what the user expects while being easy to understand is
probably important for the ease of use of Carpet, but it might be
some time until it is written.
\texttt{CarpetRegrid} contains also a routine that measures the
error, as provided in a grid function, and the automatically
decides where to refine. This is called AMR (adaptive mesh
refinement) if it works efficiently.
Much of \texttt{CarpetRegrid} is just slabbed together in an
attempt to find out what people need and expect. The thorn is a
mess, and a complete rewrite might be a good idea, once one knows
what exactly the rewritten thorn should do.
\section{Random ramblings}
Carpet uses the STL, because the STL provides very useful container
classes such as vectors, sets, and lists. Writing these abstract
datatypes oneself does not make sense in these times. It makes
much more sense to politick computer administrators to upgrade
their software.
The STL and \texttt{CarpetLib}'s classes need to be instantiated
explicitly. Several compilers have several ``automatic'' schemes
that handle all template issues ``just fine''. Except they don't.
One wants to select the following: No automatic inclusion of
\texttt{.cc} files, no automatic template instantiation at link
time. Instead, most templates are instantiated explicitly by
\texttt{CarpetLib}. It is also necessary to specify to instantiate
used templates automatically. The explicit instantiations of
\texttt{CarpetLib}'s classes live in the \texttt{.cc} files
corresponding to the \texttt{.hh} file that define the templates.
The STL templates are instantiated in the file
\texttt{CarpetLib/src/defs.cc}.
Carpet makes extensive use of the \texttt{assert()} macro in C.
This is a quick and easy way to ensure that a certain condition
holds. Assert statements abort the code if the condition does not
hold. Although I try to provide useful error messages to the user,
many unexpected cases are only caught deep inside Carpet and
manifest themselves as assertion failures. If you report an
assertion failure, it is vitally important to remember
theaccompanying file name and line number. It would also be useful
to extract from the core file a stack backtrace and the values of
the local variables of the current stack frame.
Using symmetry boundary conditions such as octant mode is currently
still awkward in Carpet. There are several reasons for this:
\texttt{CarpetRegrid} does not know about symmetries, and hence
doesn't take them into account when choosing refinement regions.
The symmetry conditions on the finer grid might be different from
the conditions on the coarser grids, and the symmetry thorns cannot
cope with this, so this situation must be avoided: one cannot use
\texttt{avoid\_origin=yes}, because the finer grids all have
\texttt{avoid\_origin=no} due to the vertex-centred refinement.
\end{document}
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