The Method of Lines thorn (version 2 - see below).

MoL provides generic integration methods for multiple thorns simultaneously. By providing a layer between the driver and evolution thorns, this should mean that some technical issues to do with mesh refinement can be ignored. It also allows you to choose different evolution methods (in time, at least). But the primary purpose is to unambiguously evolve models in different thorns at the same time.

This is version 2 - the one that will work with mesh refinement. It's a straight copy of HawkeCVS/Public/CactusMoL2/MoL2 and I haven't checked that it will work "as is" with the new name (one of the reasons it goes into Alpha).

At the moment the only evolution method guaranteed to work with mesh refinement is ICN, although RK2 should. The generic RK methods will do something that will be subtly wrong...

Note that the "old" way of registering variables (through the functions declared in header files) is still there. As soon as function aliasing settles down, this will be removed.

Also to be done:

Better documentation
Tidy up the code (especially the debugging statements)
Optimize
Add various useful time evolution methods.


git-svn-id: http://svn.cactuscode.org/arrangements/CactusNumerical/MoL/trunk@2 578cdeb0-5ea1-4b81-8215-5a3b8777ee0b

4 files changed, 2155 insertions, 0 deletions

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@@ -0,0 +1,898 @@
+\documentclass{article}
+\usepackage{graphicx}
+
+\begin{document}
+
+\title{Method of Lines}
+\author{Ian Hawke}
+\date{$ $Id$ $}
+\maketitle
+
+\abstract{The Method of Lines is a way of separating the time
+  integration from the rest of an evolution scheme. This thorn is
+  intended to take care of all the bookwork and provide some basic
+  time integration methods, allowing for easy coupling of different
+  thorns.} 
+
+\section{Purpose}
+\label{sec:purpose}
+
+The Method of Lines (MoL) converts a (system of) partial differential
+equation(s) into an ordinary differential equation containing some
+spatial differential operator. As an example, consider writing the
+hyperbolic system of PDE's
+\begin{equation}
+  \label{eq:mol1}
+  \partial_t {\bf q} + {\bf A}^i({\bf q}) \partial_i {\bf B}({\bf q})
+  = {\bf s}({\bf q})
+\end{equation}
+in the alternative form
+\begin{equation}
+  \label{eq:mol2}
+  \partial_t {\bf q} = {\bf L}({\bf q}),
+\end{equation}
+which (assuming a given discretization of space) is an ODE.
+
+Given this separation of the time and space discretizations, well
+known stable ODE integrators such as Runge-Kutta can be used to do the
+time integration. This is more modular (allowing for simple extensions
+to higher order methods), more stable (as instabilities can now only
+arise from the spatial discretization or the equations themselves) and
+also avoids the problems of retaining high orders of convergence when
+coupling different matter models.
+
+MoL can be used for hyperbolic, parabolic and even elliptic problems
+(although I definitely don't recommend the latter). As it currently
+stands it is set up for systems of equations in the first order type
+form of equation~(\ref{eq:mol2}). If you want to implement a
+multilevel scheme such as leapfrog it is not obvious to me that MoL is
+the thing to use. However if you have lots of thorns that you want to
+interact, for example ADM\_BSSN and a hydro code plus maybe EM or a
+scalar field, and they can easily be written in this sort of form,
+then you probably want to use MoL.
+
+This thorn is meant to provide a simple interface that will implement
+the MoL inside Cactus as transparently as possible. It will initially
+implement only the optimal Runge-Kutta time integrators (which are TVD
+up to RK3, so suitable for hydro) up to fourth order and iterated
+Crank Nicholson. All of the interaction with the MoL thorn should
+occur directly through the scheduler. For example, all synchronization
+steps should now be possible at the scheduler level.
+
+For more information on the Method of Lines the most comprehensive
+references are the works of Jonathan Thornburg~\cite{jt1,jt2} - see
+especially section 7.3 of the thesis. From the CFD viewpoint the
+review of ENO methods by Shu,~\cite{shu} has some information. For
+relativistic fluids the paper of Neilsen and Choptuik~\cite{nc} is
+also quite good. For Ian Hawke's viewpoint either see below or contact
+by email.
+
+\section{How to use}
+\label{sec:use}
+
+
+\subsection{Thorn users}
+\label{sec:useruse}
+
+For those who used the old version of MoL, this version is
+unfortunately slightly more effort to use. That is, for most methods
+you'll now have to set 4 parameters instead of just one. This
+documentation is still in the process of conversion...
+
+If you already have a thorn that uses the method of lines, then there
+are four main parameters that are relevant to change the integration
+method. The keyword {\tt MoL\_ODE\_Method} chooses between the
+different methods. Currently supported are {\tt RK2}, {\tt ICN} and
+{\tt Generic}. These are second order Runge-Kutta, Iterative Crank
+Nicholson, and the generic Shu-Osher type Runge-Kutta methods. To
+switch between the different types of generic methods there is also
+the keyword {\tt Generic\_Type} which is currently restricted to {\tt
+  RK} for the standard TVD Runge-Kutta methods (first to fourth order)
+and {\tt ICN} for the implementation of the Iterative Crank Nicholson
+method in generic form.
+
+The parameter {\tt MoL\_Intermediate\_Steps} controls the number of
+intermediate steps for the ODE solver. For the generic Runge-Kutta
+solvers it controls the order of accuracy of the method.  For the {\tt
+  ICN} methods this parameter controls the number of iterations taken,
+which {\bf does not check for stability}. This parameter defaults to
+3.
+
+The parameter {\tt MoL\_Num\_Scratch\_Levels} controls the amount of
+scratch space used. If this is insufficient for the method selected
+there will be an error at parameter checking time. This parameter
+defaults to 0, as no scratch space is required for the efficient ICN
+and Runge-Kutta 2 solvers. For the generic solvers this must be at
+least {\tt MoL\_Intermediate\_Steps - 1}.
+
+A final parameter is {\tt MoL\_Memory\_Always\_On} which switches on
+memory for the scratch space always if true and only during evolution
+if false. This defaults to true for speed reasons; the memory gains
+are likely to be limited unless you're doing something very memory
+intensive at initialization or analysis.
+
+\subsection{Thorn writers}
+\label{sec:writeruse}
+
+To port an existing thorn using the method of lines, or to write a new
+thorn using it, should hopefully be relatively simple. As an example,
+within the MoL arrangement is WaveMoL which duplicates the WaveToy
+thorn given by CactusWave in a form suitable for use by MoL. In this
+section, ``the thorn'' will mean the user thorn doing the physics.
+
+We start with some terminology. Grid functions are split into four
+cateogories.
+\begin{enumerate}
+\item The first are those that are evolved using a MoL form. That is,
+  a right hand side is calculated and the variable updated using
+  it. These we call {\it evolved} variables.
+\item The second category are those variables that are set by a thorn
+  at every intermediate step of the evolution, usually to respect the
+  constraints. Examples of these include the primitive variables in a
+  hydrodynamics code. Another example would be the gauge variables if
+  these were set by constraints at every intermediate step (which is
+  slightly artificial; the usual example would be the use of maximal
+  slicing, which is only applied once every $N$ complete steps). These
+  are known as {\it primitive} variables in analogy with the hydro
+  code.
+\item The third category are those variables that a thorn depends on
+  but does not set or evolve. An example would include the metric
+  terms considered from a thorn evolving matter. These are known as
+  {\it dependent} variables.
+\item The final category are those variables that do not interact with
+  MoL. These would include temporary variables for analysis or setting
+  up the initial data. These can safely be ignored.
+\end{enumerate}
+
+MoL needs to know every GF that falls in one of the first three
+groups. If a GF is evolved by one thorn but is a dependent variable in
+another (for example, the metric in full GR Hydro) then each thorn
+should register the function as they see it. For example, the hydro
+thorn will register the metric as a dependent variable and the
+spacetime thorn will register the metric as an evolved variable. The
+different variable categories are given the priority evolved,
+primitive, dependent. So if a variable is registered as belonging in
+two different categories, it is always considered by MoL to belong to
+the category with the highest priority. 
+
+Every thorn should register every grid function that it uses even if
+you expect it to be registered again by a different thorn. For
+example, a hydro thorn would register the metric variables as
+dependent, whilst the spacetime evolution thorn would register them as
+evolved (in ADM) or primitive (in ADM\_BSSN), both of which have
+precedence. To register your GFs with MoL schedule a routine in the
+bin {\tt MoL\_Register} which just contains the relevant function
+calls.  For an evolved variable the GF corresponding to the update
+term (${\bf L}({\bf q})$ in equation~(\ref{eq:mol2})) should be
+registered at the same time, using the call {\tt MoL\_RegisterVar($q,
+  L(q)$)}. For the primitive and dependent variables the function
+calls are {\tt MoL\_RegisterPrimitive($q$)} and {\tt
+  MoL\_RegisterDepends($q$)} respectively. Here $q$ is the {\it
+  variable index} of the GF you wish to register, and $L(q)$ the
+variable index of the associated right hand side GF. These are found
+using the {\tt CCTK\_VarIndex*} functions.
+
+There are both C and Fortran versions of these functions; see the
+reference in section~\ref{sec:molfns}. For the C calls the prototypes
+are given in the file {\tt MoL.h} which can be included without hard
+paths by adding the line
+
+{\tt USES INCLUDE: MoL.h}
+
+\noindent to the {\tt interface.ccl} file of your thorn.
+
+{\bf UPDATE} This has now been switched to using function
+aliasing. The include file still exists, but is commented out in the
+{\tt interface.ccl}. For an example of how to use the aliased
+functions, see thorn WaveMoL.
+
+Having done that, one routine (or group of routines) which we'll here
+call {\tt Thorn\_CalcRHS} must be defined. This does all the finite
+differencing that you'd usually do, applied to ${\bf q}$, and finds
+the right hand sides which are stored in ${\bf L}$. This routine
+should be scheduled in {\tt MoL\_CalcRHS}. The precise order that
+these are scheduled should not matter, because no updating of any of
+the user thorns ${\bf q}$ will be done until after all the RHSs are
+calculated. {\bf Important note:} all the finite differencing must be
+applied to the most recent time level ${\bf q}$ and not to the
+previous time level ${\bf q}_p$ as you would normally do. Don't worry
+about setting up the data before the calculation, as MoL will do that
+automatically.
+
+Finally, if you have some things that have to be done after each
+update to an intermediate level, these should be scheduled in {\tt
+  MoL\_PostStep}. Examples of things that need to go here include the
+recalculation of primitive variables for hydro codes, the application
+of boundary conditions\footnote{It is possible to alter the
+  calculation of {\bf L} so that boundary conditions are automatically
+  updated and do not need setting. This is slightly tricksy. For an
+  example of how this would work see the new radiative boundary
+  condition in ADM\_BSSN. For more on this see section 7.3.4
+  of~\cite{jt1}.}, the solution of elliptic equations (although this
+would be a very expensive place to solve them, some sets of equations
+might require the updating of some variables by constraints in this
+fashion). When applying boundary conditions the cleanest thing to do
+is to write a routine applying the symmetries to the appropriate GFs
+and, when calling it from the scheduler, adding the {\tt SYNC}
+statement to the appropriate groups. An example is given by the
+routine {\tt WaveToyMoL\_Boundaries} in thorn WaveMoL.
+
+Points to note. The thorn routine {\tt Thorn\_CalcRHS} does not need
+to know and in fact should definitely not know where precisely in the
+MoL step it is. It just needs to know that it is receiving {\it some}
+intermediate data stored in the GFs ${\bf q}$ and that it should
+return the RHS ${\bf L}({\bf q})$. All the book-keeping to ensure that
+it is passed the correct intermediate state at that the GFs contain
+the correct data at the end of the MoL step will be dealt with by the
+MoL thorn and the flesh. Also the synchronization of grids across
+separate processors will be dealt with by the MoL thorn and the flesh.
+
+Final point. Currently this MoL thorn is not guaranteed to work with
+mesh refinement codes. Aside from a number of little details that need
+to be worked out, currently MoL does not ensure that {\tt
+  cctk\_delta\_time} is correctly set in the intermediate stages of
+the calculation. This will hopefully change, but for the immediate
+future don't expect this to work with any mesh refinement code.
+
+Update. It is now possible to change the category of a GF in the
+middle of a run. This was added solely to allow for mixed slicing for
+Einstein. In this case the lapse is an evolved function if using
+$1+\log$ slicing, a primitive variable if using maximal or static
+slicing, and a dependent variable if evolved in some unknown
+manner. Currently there exist functions for changing to any of the
+four categories listed above.
+
+{\bf I strongly suggest that these functions are not used}. They are
+almost completely untested, and even those that are tested are far
+from stable. Also, when moving to the version of this thorn that
+supports Mesh Refinement it is quite possible that these functions may
+be dropped.
+
+
+\section{Example}
+\label{sec:example}
+
+As a fairly extended example of how to use MoL I'll outline how
+ADM\_BSSN works in this context. The actual implementation of this is
+given in the thorn {\tt AEIDevelopment/BSSN\_MoL}. 
+
+As normal the required variables are defined in the {\tt
+  interface.ccl} file, together with the associated source terms. For
+example, the conformal factor and source are defined by
+
+\begin{verbatim}
+real ADM_BSSN_phi type=GF timelevels=2
+{
+  ADM_BS_phi
+} "ADM_BSSN_phi"
+
+real ADM_BSSN_sources type=GF
+{
+...,
+  adm_bs_sphi,
+...
+}
+\end{verbatim}
+Also in this file we write {\tt USES INCLUDE: MoL.h}.
+
+Once the sources are defined the registration with MoL is required,
+for which the essential file is {\tt MoLRegister.c}. As we're
+registering from C we ensure that we {\tt \#include "MoL.h"}. In the
+ADM\_BSSN system the standard metric coefficients $g_{ij}$ are not
+evolved, and neither are the standard extrinsic curvature components
+$K_{ij}$.  However these are used by ADM\_BSSN in a number of places,
+and are calculated from evolved quantities at the appropriate points.
+In the MoL terminology these variables are {\it primitive}. As the
+appropriate storage is defined in thorn Einstein, the actual calls
+have the form
+
+\begin{verbatim}
+ ierr += MoL_RegisterPrimitive(CCTK_VarIndex("einstein::kxx"));
+\end{verbatim}
+
+\noindent The actual evolved variables include things such as the
+conformal factor. This, and the appropriate source term, is defined in
+thorn ADM\_BSSN, and so the call has the form
+
+\begin{verbatim} 
+ ierr += MoL_RegisterVar(CCTK_VarIndex("adm_bssn::ADM_BS_phi"),
+                         CCTK_VarIndex("adm_bssn::adm_bs_sphi")); 
+\end{verbatim}
+
+
+As well as the evolved variables, and those constrained variables such
+as the metric, there are the gauge variables. Precisely what status
+these have depends on how they are set. If harmonic or 1+log slicing
+is used then the lapse is evolved:
+
+\begin{verbatim}
+ ierr += MoL_RegisterVar(CCTK_VarIndex("einstein::alp"),
+                         CCTK_VarIndex("adm_bssn::adm_bs_salp")); 
+\end{verbatim}
+
+\noindent If maximal or static slicing is used then the lapse is a
+primitive variable\footnote{Note that this is actually a bit of a
+  hack. The rational for {\it dependent} variables was to deal with
+  maximal slicing. However it turned out that I hadn't thought it
+  through correctly and that the treatment for primitive variables was
+  required.}:
+
+\begin{verbatim}
+ ierr += MoL_RegisterPrimitive(CCTK_VarIndex("einstein::alp"));
+\end{verbatim}
+
+\noindent Finally, if none of the above apply we assume that the lapse
+is evolved in some unknown fashion, and so it must be registered as a
+dependent variable:
+
+\begin{verbatim}
+ ierr += MoL_RegisterDepends(CCTK_VarIndex("einstein::alp"));
+\end{verbatim}
+
+However, it is perfectly possible that we may wish to change how we
+deal with the gauge during the evolution. This is dealt with in the
+file {\tt PreLoop.F}. If the slicing changes then the appropriate
+routine is called. For example, if we want to use 1+log evolution then
+we call 
+
+\begin{verbatim}
+ call CCTK_VarIndex(lapseindex,"einstein::alp")
+ call CCTK_VarIndex(lapserhsindex,"adm_bssn::adm_bs_salp")
+ call MoL_ChangeVarToEvolved(ierr, lapseindex, lapserhsindex)
+\end{verbatim}
+
+\noindent It is not required to tell MoL what the lapse is changing
+{\it from}, or indeed if it is changing at all; MoL will work this out
+for itself.
+
+Finally there are the routines that we wish to apply after every
+intermediate step. These are {\tt ADM\_BSSN\_removetrA} which enforces
+various constraints (such as the tracefree conformal extrinsic
+curvature remaining trace free), {\tt ADM\_BSSN\_Boundaries} which
+applies symmetry boundary conditions as well as various others (such
+as some of the radiative boundary conditions). Note all the calls to
+{\tt SYNC} at this point. We also convert from the updated BSSN
+variables back to the standard ADM variables in {\tt
+  ADM\_BSSN\_StandardVariables}, and also update the time derivative
+of the lapse in {\tt ADM\_BSSN\_LapseChange}.
+
+\section{The gory details}
+\label{sec:gory}
+
+This section is more for people wishing to maintain and extend the
+code. {\bf NEEDS TOTAL REWRITE FOR MOL2}
+
+There are three essential parts to the MoL thorn. The first is the
+registration process where the MoL thorn is told just how many GFs it
+should be looking to integrate, and where they and their RHSs
+are. This is split into three separate routines.
+\begin{itemize}
+\item {\tt MoL\_Init\_Register} Called by the MoL thorn at {\tt
+    postinitial}.  Initializes the number of variables to the maximum
+  possible number (given by {\tt CCTK\_NumVars()}). Sets up two arrays
+  of pointers to GFs (CCTK\_Real**), one for the data ({\tt qindex})
+  and one for the RHSs ({\tt qrhsindex}). It also sets up index arrays
+  for the primitive and dependent variables.
+\item {\tt MoL\_Register} This group lets MoL knows where all the data
+  is and should contain all the user thorn calls to {\tt
+    MoL\_RegisterVar()}. This just associates the index of the
+  GFs to be evolved with their RHSs, and increments the number of
+  variables to be evolved. Also contains the routines to register the
+  primitive and dependent variables.
+\item {\tt MoL\_End\_Register} This could be used to reallocate the
+  index arrays to the correct size. For the moment I've decided that
+  the tiny memory gain is outweighed by the possiblity of problems
+  using realloc.
+\end{itemize}
+
+The second part of the MoL thorn sets up all the scratch space and the
+parameters for the generalized RK routines. This is also scheduled at
+postinitial. This contains the routines
+
+\begin{itemize}
+\item {\tt MoL\_SetupScratch} Sets up the scratch space. Sets the
+  scalar {\tt mol\_num\_scratch} which is the size of the required
+  scratch space from the defaults, then allocates the scratch arrays. 
+\item {\tt MoL\_RKSetup} Sets up the $\alpha$ and $\beta$ arrays for
+  the generalized Runge Kutta step. Note that as we're currently only
+  storing the most recent RHS arrays the $\beta$ arrays are single
+  index only.
+\end{itemize}
+
+The final part of the MoL thorn is the evolution step. This is again
+split into a number of separate routines and groups. Firstly we
+describe a single evolution step, and what MoL expects.
+
+By a single evolution step we here mean the execution of everything
+inside the schedule bin {\tt EVOL}. MoL expects that every GF that it
+knows about is allocated, initialized and lives at the same instant of
+computational time. MoL also expects the ghost zones and boundaries to
+be set correctly. Wherever possible the first two are checked. MoL
+expects that the driver has rotated the timelevels so that the last
+set of complete, consistent data is stored in ${\bf q}\_p$. MoL also
+knows that the dependent variables may or may not have been evolved by
+the time the MoL evolution group is executed.
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=3cm]{MoLdia1}
+\includegraphics[width=3cm]{MoLdia2}
+\includegraphics[width=3cm]{MoLdia3}
+\end{center}
+\caption{How MoL treats the three different types of variables. The
+  MoL step is performed at evolution after the driver has rotated the
+  timelevels, which occurs right at the start of every {\tt
+    CCTK\_EVOL} step. The physics thorns expect the most recent data
+  to be at the current time level (the top solid line). So the first
+  step for most types of variable is to copy or pointer switch the
+  data from the previous time level (bottom solid line) where the most
+  recent data exists after rotation. Figure (a) shows the {\it
+    evolved} variables.  At each intermediate step the data is updated
+  into the current time level and, if necessary, stored in scratch
+  space.  Figure (b) shows the {\it primitive} variables. As these are
+  set by the physics thorns into the current time level all that is
+  required is the initial copy. Figure (c) shows the {\it dependent}
+  variables for which we cannot know whether these variables are
+  evolved before or after MoL at evolution, and hence whether the
+  current data is already filled before MoL starts, we must store the
+  current data in scratch space first, then do the copy from the
+  previous level, and then at the end of the MoL step return the data
+  to the initial state.}
+\label{MoLvariables}
+\end{figure}
+
+MoL plays around with the timelevels during the intermediate
+steps. This is required so that the latest level in the MoL evolution
+is always stored in the same place so that the user thorns can easily
+access them. This place is the current time level ${\bf q}$. This
+ensures that MoL works correctly with all the standard boundary
+condition routines and minimizes the effort of porting non timelevel
+aware thorns (I hope!).
+
+An outline of the schedule is as follows:
+\begin{equation}
+  \label{eq:schedoutline}
+  \begin{array}[l]{l}
+    \texttt{MoL\_StartStep} \\
+    \texttt{MoL\_Step WHILE counter \{} \\
+    \begin{array}[l]{l}
+      \texttt{MoL\_CalcRHS \{ \}} \\
+      \texttt{MoL\_Add} \\
+      \texttt{MoL\_PostStep \{ \}} \\
+    \end{array} \\
+    \texttt{\}} \\
+    \texttt{MoL\_EndStep \{ \}}
+  \end{array}
+\end{equation}
+
+Each different type of variable is treated slightly differently by
+MoL. Each is assumed to have at least 2 timelevels (although this is
+checked, a fatal error occurs otherwise). Before entering the loop
+over the intermediate steps MoL will first copy (pointer switch?) the
+data into the current time levels. Before doing this the current
+timelevel of any {\it dependent} variables is copied to scratch space,
+as it may have already been updated. During the loop over the
+intermediate steps only the {\it evolved} variables are directly
+altered by MoL. When all user thorns have given their right hand side
+GFs the evolved variables are updated into the current time level.
+This data may also be copied to scratch space if required for later.
+The primitive variables are assumed to be set by the user thorn,
+either in the calculation of the RHS or during {\tt MoL\_PostStep} at
+the end of each intermediate step. The dependent variables are assumed
+to remain completely unchanged. After all the intermediate steps the
+data in the current and previous timelevels is ``correct'' and
+consistent for the evolved and the temporary variables. The data for
+the current timelevel for the dependent variables is recovered from
+scratch space. Precisely how the different variables are
+treated is shown in figure~\ref{MoLvariables}.
+
+\begin{itemize}
+\item {\tt MoL\_StartStep} This ensures that the integer keeping
+    track of where we are in the MoL step is set to the correct
+    value. It also copies the previous data ${\bf q}_p$ into the
+    current position ${\bf q}$ ready for the first step.
+\item {\tt MoL\_Step} This is the main part of the thorn. The scheduler
+  allows us to loop over this group the correct number of times to
+  complete a single Cactus evolution step. Contained within this group
+  are:
+  \begin{itemize}
+  \item {\tt MoL\_CalcRHS} The schedule group within which the user
+    thorns will schedule their routines. These routines should
+    calculate the GFs ${\bf L}({\bf q})$. The order these routines are
+    scheduled within this group should not matter.
+  \item {\tt MoL\_Add} This step performs the time integration
+    depending on where in the MoL step we are. Updates directly into
+    the current GF and copies to the scratch space if required.
+  \item {\tt MoL\_PostStep} Another schedule group within which the
+    user thorns can schedule such things as primitive variable
+    recovery, boundary conditions, etc.
+  \item {\tt MoL\_End} Just alters the scalar tracking the position in
+    the MoL loop.
+  \end{itemize}
+\end{itemize}
+
+Finally, there are the routines {\tt MoL\_FreeScratch} and {\tt
+  MoL\_RKFree} which free up the memory that was explicitly taken. For
+the moment these routines are scheduled at postevol.
+
+
+\section{Adding new numerical integrators}
+\label{sec:newmeth}
+
+There are two obvious ways of adding new ODE integrators into MoL. The
+first is to follow the route used in the efficient RK2 and ICN
+methods. That is, you let the underlying infrastructure define the
+scratch space but you do all the addition yourself. It's probably best
+if you model your integrator on one of the efficient routines to start
+with. 
+
+The alternative is to use the generic integrator. To use this you just
+need to add the correct set of $\alpha$ and $\beta$ coefficients so
+that the generic routine can perform the additions. You'll also have
+to set up the keyword parameters and so on.
+
+\section{To do list}
+\label{sec:todo}
+
+\begin{itemize}
+\item The documentation must be improved, especially inside the code
+  itself. 
+\item Errors are currently not handled well (if at all). This must be
+  fixed.
+\item A test suite is required. I'm not sure how to do this without
+  using WaveMoL, but we could try.
+\item In order to make the code work with a Mesh Refinement driver,
+  the scratch spaces must be changed to be grid functions. This is
+  currently impossible, but Tom Goodale will add the appropriate bits
+  to the flesh to make it possible to do. At that point the entire
+  code will probably be rewritten.
+\end{itemize}
+
+\section{Functions provided by MoL}
+\label{sec:molfns}
+
+{\bf Note for Cactus people: I'd really like there to be a generic
+  style file so that I could use the same function environments as in
+  the User's Guide. It would also be nice if the parsing of the
+  interface files recognized function aliasing when that appears, but
+  that's a long distant wish}.
+
+All the functions listed below return error codes in theory. However
+at this current point in time they always return 0 (success). Any
+failure to register or change a GF is assumed fatal and MoL will
+issue a level 0 warning stopping the code. This may change in future,
+in which case negative return values will indicate errors.
+
+\subsection*{MoL\_RegisterVar}
+
+Tells MoL that the given GF is in the evolved category with the
+associated update GF. 
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_RegisterVar(CCTK\_INT varindex,
+  CCTK\_INT rhsindex) \\
+  & {\bf Fortran} & MoL\_RegisterVar(CCTK\_INT ierr, CCTK\_INT varindex,
+  CCTK\_INT rhsindex) 
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the GF to be evolved \\
+  & {\bf rhsindex} & - & Index of the associated update GF.
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_Register}.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_RegisterVar(CCTK\_VarIndex("wavetoymol::phi"), \\
+  && \qquad CCTK\_VarIndex("wavetoymol::phirhs")); \\ 
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``wavetoymol::phi'')
+  \\
+  && call CCTK\_VarIndex(rhsindex, ``wavetoymol::phirhs'') \\
+  && call MoL\_RegisterVar(ierr, varindex, rhsindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_RegisterDepends, MoL\_RegisterPrimitive, \\
+  & MoL\_ChangeVarToEvolved.
+\end{tabular}
+
+
+\subsection*{MoL\_RegisterDepends}
+
+Tells MoL that the given GF is in the dependent category.
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_RegisterDepends(CCTK\_INT varindex) \\
+  & {\bf Fortran} & MoL\_RegisterDepends(CCTK\_INT ierr, CCTK\_INT varindex) 
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the dependent GF.
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_Register}.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_RegisterDepends(CCTK\_VarIndex("einstein::alp")); \\ 
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``einstein::alp'')
+  \\
+  && call MoL\_RegisterDepends(ierr, varindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_RegisterVar, MoL\_RegisterPrimitive, \\
+  & MoL\_ChangeVarToDependent.
+\end{tabular}
+
+
+\subsection*{MoL\_RegisterPrimitive}
+
+Tells MoL that the given GF is in the primitive category.
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_RegisterPrimitive(CCTK\_INT varindex) \\
+  & {\bf Fortran} & MoL\_RegisterPrimitive(CCTK\_INT ierr, CCTK\_INT varindex) 
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the primitive GF.
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_Register}.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_RegisterPrimitive(CCTK\_VarIndex("einstein::alp")); \\ 
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``einstein::alp'')
+  \\
+  && call MoL\_RegisterPrimitive(ierr, varindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_RegisterVar, MoL\_RegisterDepends, \\
+  & MoL\_ChangeVarToPrimitive.
+\end{tabular}
+
+
+\subsection*{MoL\_ChangeVarToEvolved}
+
+Sets a GF to belong to the evolved category, with the associated
+update GF. Not used for the initial setting.
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_ChangeVarToEvolved(CCTK\_INT varindex, CCTK\_INT
+  rhsindex) \\ 
+  & {\bf Fortran} & MoL\_ChangeVarToEvolved(CCTK\_INT ierr, CCTK\_INT varindex,
+  CCTK\_INT rhsindex)  
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the GF to be evolved. \\
+  & {\bf rhsindex} & - & Index of the associated update GF.
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_PreStep}. \\
+  & Note that this function was designed to allow mixed slicings for
+  thorn Einstein. \\
+  & This set of functions is largely untested and should be used with
+  great care.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_ChangeVarToEvolved(CCTK\_VarIndex("einstein::alp"),\\
+  && \qquad CCTK\_VarIndex(``adm\_bssn::adm\_bs\_salp'')); \\
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``einstein::alp'') \\
+  && call CCTK\_VarIndex(rhsindex, ``adm\_bssn::adm\_bs\_salp'') \\
+  && call MoL\_ChangeVarToEvolved(ierr, varindex, rhsindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_RegisterVar, MoL\_ChangeVarToDependent, \\
+  & MoL\_ChangeVarToPrimitive, MoL\_ChangeVarToNone.
+\end{tabular}
+
+
+\subsection*{MoL\_ChangeVarToDependent}
+
+Sets a GF to belong to the dependent category. Not used for the
+initial setting. 
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_ChangeVarToDependent(CCTK\_INT varindex) \\ 
+  & {\bf Fortran} & MoL\_ChangeVarToDependent(CCTK\_INT ierr, CCTK\_INT varindex)  
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the dependent GF.
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_PreStep}. \\
+  & Note that this function was designed to allow mixed slicings for
+  thorn Einstein. \\
+  & This set of functions is largely untested and should be used with
+  great care.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_ChangeVarToDependent(CCTK\_VarIndex("einstein::alp")); \\
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``einstein::alp'') \\
+  && call MoL\_ChangeVarToDependent(ierr, varindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_RegisterDepends, MoL\_ChangeVarToEvolved, \\
+  & MoL\_ChangeVarToPrimitive, MoL\_ChangeVarToNone.
+\end{tabular}
+
+\subsection*{MoL\_ChangeVarToPrimitive}
+
+Sets a GF to belong to the primitive category. Not used for the
+initial setting. 
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_ChangeVarToPrimitive(CCTK\_INT varindex) \\ 
+  & {\bf Fortran} & MoL\_ChangeVarToPrimitive(CCTK\_INT ierr, CCTK\_INT varindex)  
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the primitive GF.
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_PreStep}. \\
+  & Note that this function was designed to allow mixed slicings for
+  thorn Einstein. \\
+  & This set of functions is largely untested and should be used with
+  great care.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_ChangeVarToPrimitive(CCTK\_VarIndex("einstein::alp")); \\
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``einstein::alp'') \\
+  && call MoL\_ChangeVarToPrimitive(ierr, varindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_RegisterPrimitive, MoL\_ChangeVarToEvolved, \\
+  & MoL\_ChangeVarToDependent, MoL\_ChangeVarToNone.
+\end{tabular}
+
+\subsection*{MoL\_ChangeVarToNone}
+
+Sets a GF to belong to the ``unknown'' category. Not used for the
+initial setting. 
+
+\vskip 3mm
+
+\begin{tabular}[l]{l l l}
+  {\bf Synopsis} && \\
+  & {\bf C} & CCTK\_INT ierr = MoL\_ChangeVarToNone(CCTK\_INT varindex) \\ 
+  & {\bf Fortran} & MoL\_ChangeVarToNone(CCTK\_INT ierr, CCTK\_INT varindex)  
+\end{tabular}
+
+\begin{tabular}[l]{l l c l}
+  {\bf Arguments} &&& \\
+  & {\bf varindex} & - & Index of the GF to be unset. 
+\end{tabular}
+
+\begin{tabular}[l]{l l}
+  {\bf Discussion} & \\
+  & Should be called in a function scheduled in {\tt MoL\_PreStep}. \\
+  & Note that this function was designed to allow mixed slicings for
+  thorn Einstein. \\
+  & This set of functions is largely untested and should be used with
+  great care.
+\end{tabular}
+
+\begin{tabular}[l]{l l l}
+  {\bf Examples} && \\
+  & {\bf C} & ierr =
+  MoL\_ChangeVarToNone(CCTK\_VarIndex("einstein::alp")); \\
+  & {\bf Fortran} & call CCTK\_VarIndex(varindex, ``einstein::alp'') \\
+  && call MoL\_RegisterNone(ierr, varindex)
+\end{tabular}
+
+\begin{tabular}[l]{l l }
+  {\bf See Also} & \\
+  & CCTK\_VarIndex, MoL\_ChangeVarToEvolved, MoL\_ChangeVarToDependent, \\
+  & MoL\_ChangeVarToPrimitive.
+\end{tabular}
+
+
+
+
+\begin{thebibliography}{1}
+
+\bibitem{jt1}
+J. Thornburg.
+\newblock {N}umerical {R}elativity in {B}lack {H}ole {S}pacetimes. 
+\newblock Unpublished thesis, University of British Columbia.
+\newblock 1993.
+\newblock Available from \mbox{\tt
+  http://www.aei.mpg.de/\~{}jthorn/phd/html/phd.html}. 
+
+\bibitem{jt2}
+J. Thornburg.
+\newblock A {3+1} {C}omputational {S}cheme for {D}ynamic {S}pherically
+{S}ymmetric {B}lack {H}ole {S}pacetimes -- {II}: {T}ime {E}volution.
+\newblock Preprint {\tt gr-qc/9906022}, submitted to {\em Phys. Rev.}
+{\bf D}. 
+
+\bibitem{shu}
+C. Shu.
+\newblock {H}igh {O}rder {ENO} and {WENO} {S}chemes for
+{C}omputational {F}luid {D}ynamics.
+\newblock In T.~J. Barth and H. Deconinck, editors {\em High-Order
+  Methods for Computational Physics}. Springer, 1999.
+\newblock A related online version can be found under {\em Essentially
+  {N}on-{O}scillatory and {W}eighted {E}ssentially {N}on-{O}scillatory
+  {S}chemes for {H}yperbolic {C}onservation {L}aws} at {\tt
+  http://www.icase.edu/library/reports/rdp/97/97-65RDP.tex.refer.html}. 
+
+\bibitem{nc}
+D.~W. Neilsen and M.~W. Choptuik.
+\newblock Ultrarelativistic fluid dynamics.
+\newblock {\em Class. Quantum Grav.}, {\bf 17}:\penalty0 733--759, 2000.
+
+\end{thebibliography}
+
+\include{interface}
+\include{param}
+\include{schedule}
+
+\end{document}