## Method of Lines

Date

### 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.

### 1 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 {CactusBase_MoL_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 {CactusBase_MoL_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 physical 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 (??). 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 schedule level. This is essential for interacting cleanly with different drivers, especially to make Mesh Refinement work.

For more information on the Method of Lines the most comprehensive references are the works of Jonathan Thornburg [12] - see especially section 7.3 of the thesis. From the CFD viewpoint the review of ENO methods by Shu, [3], has some information. For relativistic fluids the paper of Neilsen and Choptuik [4] is also quite good.

### 2 How to use

#### 2.1 Thorn users

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.

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 MoL_ODE_Method chooses between the different methods. Currently supported are RK2, RK3, ICN, ICN-Avg and Generic. These are second order Runge-Kutta, third order Runge-Kutta, Iterative Crank Nicholson, Iterative Crank Nicholson with averaging, and the generic Shu-Osher type Runge-Kutta methods. To switch between the different types of generic methods there is also the keyword Generic_Type which is currently restricted to RK for the standard TVD Runge-Kutta methods (first to fourth order) and ICN for the implementation of the Iterative Crank Nicholson method in generic form.

Full descriptions of the currently implemented methods are given in section 4.

The parameter 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 ICN methods this parameter controls the number of iterations taken, which does not check for stability. This parameter defaults to 3.

The parameter 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 and 3 solvers. For the generic solvers this must be at least MoL_Intermediate_Steps - 1.

Another parameter is 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.

There is also a parameter MoL_NaN_Check that will check your RHS grid functions for NaNs using the NaNChecker thorn from CactusUtils. This will make certain that you find the exact grid function computing the first NaN; of course, this may not be the real source of your problem.

The parameter disable_prolongation only does anything if you are using mesh refinement, and in particular Carpet. With mesh refinement it may be necessary to disable prolongation in intermediate steps of MoL. This occurs when evolving systems containing second spatial derivatives. This is done by default in MoL. If your system is purely first order in space and time you may wish to set this to "no".

Ideally, initial data thorns should always set initial data at all time levels. However, sometimes initial data thorns fail to do this. In this case you can do one of three things:

• Fix the initial data thorn. This is the best solution.

• If you’re using Carpet, it has some facilities to take forward/backward time steps to initialize multiple time levels. See the Carpet parameters init_each_timelevel and init_3_timelevels for details.

• Finally, if you set (the MoL parameter) initial_data_is_crap, MoL will copy the current time level of all variables it knows about (more precisely, using the terminology of section 2.2, all evolved, save-and-restore, and constrained variables which have multiple time levels) to all the past time levels. Note that this copies the same data to each past time level; this will be wrong if your spacetime is time-dependent!

If enabled, the copy happens in the CCTK_POSTINITIAL schedule bin. By default this happens before the MoL_PostStep schedule group; the parameter copy_ID_after_MoL_PostStep can be used to change this to after MoL_PostStep.

#### 2.2 Thorn writers

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.

1. 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 evolved variables.

2. 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 constrained variables.

3. 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. Due to the way that MoL deals with these, they are known as Save and Restore variables.

4. 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.

As a generic rule of thumb, variables for which you have a time evolution equation are evolved (obviously), variables which your thorn sets but does not evolve are constrained, and any other variables which your thorn reads during evolution is a Save and Restore variable.

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 constrained 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 Save and Restore variable and the spacetime thorn will register the metric as an evolved variable. The different variable categories are given the priority evolved, constrained, Save and Restore. 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.

MoL needs to know the total number of GFs in each category at parameter time. To do this, your thorn needs to use some accumulator parameters from MoL. As an example, here are the paramaters from WaveMoL:

shares: MethodOfLines

USES CCTK_INT MoL_Num_Evolved_Vars
USES CCTK_INT MoL_Num_Constrained_Vars
USES CCTK_INT MoL_Num_SaveAndRestore_Vars

restricted:

CCTK_INT WaveMoL_MaxNumEvolvedVars \
"The maximum number of evolved variables used by WaveMoL" \
ACCUMULATOR-BASE=MethodofLines::MoL_Num_Evolved_Vars
{
5:5           :: "Just 5: phi and the four derivatives"
} 5

CCTK_INT WaveMoL_MaxNumConstrainedVars \
"The maximum number of constrained variables used by WaveMoL" \
ACCUMULATOR-BASE=MethodofLines::MoL_Num_Constrained_Vars
{
0:1           :: "A small range, depending on testing or not"
} 1

CCTK_INT WaveMoL_MaxNumSandRVars \
"The maximum number of save and restore variables used by WaveMoL" \
ACCUMULATOR-BASE=MethodofLines::MoL_Num_SaveAndRestore_Vars
{
0:1           :: "A small range, depending on testing or not"
} 1


This should give the maximum number of variables that your thorn will register.

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 Save and Restore, whilst the spacetime evolution thorn would register them as evolved (in ADM) or constrained (in ADM_BSSN), both of which have precedence. To register your GFs with MoL schedule a routine in the bin 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 (??)) should be registered at the same time. The appropriate functions are given in section 5.

These functions are provided using function aliasing. For details on using function aliasing, see sections B10.5 and F2.2.3 of the UsersGuide. For the case of real GFs, you simply add the following lines to your interface.ccl:

##########################################
### PROTOTYPES - DELETE AS APPLICABLE! ###
##########################################

CCTK_INT FUNCTION MoLRegisterEvolved(CCTK_INT EvolvedIndex, CCTK_INT RHSIndex)
CCTK_INT FUNCTION MoLRegisterEvolvedSlow(CCTK_INT EvolvedIndex, CCTK_INT RHSIndex)
CCTK_INT FUNCTION MoLRegisterConstrained(CCTK_INT ConstrainedIndex)
CCTK_INT FUNCTION MoLRegisterSaveAndRestore(CCTK_INT SandRIndex)
CCTK_INT FUNCTION MoLRegisterEvolvedGroup(CCTK_INT EvolvedIndex, \
CCTK_INT RHSIndex)
CCTK_INT FUNCTION MoLRegisterConstrainedGroup(CCTK_INT ConstrainedIndex)
CCTK_INT FUNCTION MoLRegisterSaveAndRestoreGroup(CCTK_INT SandRIndex)
CCTK_INT FUNCTION MoLChangeToEvolved(CCTK_INT EvolvedIndex, CCTK_INT RHSIndex)
CCTK_INT FUNCTION MoLChangeToEvolvedSlow(CCTK_INT EvolvedIndex, CCTK_INT RHSIndex)
CCTK_INT FUNCTION MoLChangeToConstrained(CCTK_INT ConstrainedIndex)
CCTK_INT FUNCTION MoLChangeToSaveAndRestore(CCTK_INT SandRIndex)
CCTK_INT FUNCTION MoLChangeToNone(CCTK_INT RemoveIndex)

#############################################
### USE STATEMENT - DELETE AS APPLICABLE! ###
#############################################

USES FUNCTION MoLRegisterEvolved
USES FUNCTION MoLRegisterEvolvedSlow
USES FUNCTION MoLRegisterConstrained
USES FUNCTION MoLRegisterSaveAndRestore
USES FUNCTION MoLRegisterEvolvedGroup
USES FUNCTION MoLRegisterConstrainedGroup
USES FUNCTION MoLRegisterSaveAndRestoreGroup
USES FUNCTION MoLChangeToEvolved
USES FUNCTION MoLChangeToConstrained
USES FUNCTION MoLChangeToSaveAndRestore
USES FUNCTION MoLChangeToNone


Note that the list of parameters not complete; see the section on parameters for the use of arrays. However, the list of functions is, and is expanded on in section 5. MoL will check whether a group or variable is a GF or an array and whether it is real.

Having done that, one routine (or group of routines) which we’ll here call 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 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. 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 MoL_PostStep. Examples of things that need to go here include the recalculation of primitive variables for hydro codes, the application of boundary conditions1 , 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 SYNC statement to the appropriate groups. An example is given by the routine WaveToyMoL_Boundaries in thorn WaveMoL.

Points to note. The thorn routine 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 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.

When using a multirate scheme the thorn routine Thorn_CalcRHS must check the MoL grid scalars MoL::MoL_SlowStep and MoL::MoL_SlowPostStep which MoL sets to true (non-zero) if it is time for the slow sector to compute its RHS or to apply its poststep routine. MoL::MoL_SlowPostStep is always true outside of the RHS loop, eg. in CCTK_POST_RESTRICT.

#### 2.3 Evolution method writers

If you want to try adding a new evolution method to MoL the simplest way is to use the generic table option to specify it completely in the parameter file - no coding is required at all.

All the generic methods evolve the equation \begin {equation} \label {CactusBase_MoL_eq:mol3} \partial _t {\bf q} = {\bf L}({\bf q}) \end {equation} using the following algorithm for an $$N$$-step method:

\begin {eqnarray} \label {CactusBase_MoL_eq:genrk1-scheme} {\bf q}^{(0)} & = & {\bf q}^n, \nonumber \\ {\bf q}^{(i)} & = & \sum _{k=0}^{i-1} \left ( \alpha _{ik} {\bf q}^{(k)} \right ) + \Delta t \beta _{i-1} {\bf L} ( {\bf q}^{(i-1)} ), \qquad i = 1, \dots , N, \\ {\bf q}^{n+1} & = & {\bf q}^{(N)}. \nonumber \end {eqnarray}

This method is completely specified by $$N$$ (GenericIntermediateSteps) and the $$\alpha$$ (GenericAlphaCoeffs) and $$\beta$$ (GenericBetaCoeffs) arrays. The names in parentheses give the keys in a table that MoL will use. This table is created from the string parameter Generic_Method_Descriptor.

As an example, the standard TVD RK2 method that is implemented both in optimized and generic form is written as

\begin {eqnarray} \label {CactusBase_MoL_eq:rk2-scheme} {\bf q}^{(1)} & = & {\bf q}^n + \Delta t {\bf L} ({\bf q}^n), \\ {\bf q}^{n+1} & = & \frac {1}{2} \left ( {\bf q}^n + {\bf q}^{(1)} + \Delta t {\bf L} ({\bf q}^{(1)}) \right ). \end {eqnarray}

To implement this using the generic table options, use

methodoflines::MoL_Intermediate_Steps = 2
methodoflines::MoL_Num_Scratch_Levels = 1
methodoflines::Generic_Method_Descriptor = \
"GenericIntermediateSteps = 2 \
GenericAlphaCoeffs = { 1.0 0.0 0.5 0.5 } \
GenericBetaCoeffs = { 1.0 0.5 }"


The number of steps specified in the table must be the same as MoL_Intermediate_Steps, and the number of scratch levels should be at least MoL_Intermediate_Steps - 1.

The generic methods are somewhat inefficient for use in production runs, so it is frequently better to write an optimized version once you are happy with the method. To do this you should

• write your code into a new file, called from the scheduler under the alias MoL_Add,

• make certain that at each intermediate step the correct values of cctk_time and cctk_delta_time are set in SetTime.c for mesh refinement, boundary conditions and so on,

• make certain that you check for the number of intermediate steps in ParamCheck.c.

### 3 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 AEIThorns/BSSN_MoL.

As normal the required variables are defined in the interface.ccl file, together with the associated source terms. For example, the conformal factor and source are defined by

real ADM_BSSN_phi type=GF timelevels=2
{

{
...,
...
}


Also in this file we write the function aliasing prototypes.

Once the sources are defined the registration with MoL is required, for which the essential file is MoLRegister.c. 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 constrained. As the appropriate storage is defined in thorn ADMBase, the actual calls have the form

 ierr += MoLRegisterConstrained(CCTK_VarIndex("ADMBase::kxx"));


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

 ierr += MoLRegisterEvolved(CCTK_VarIndex("adm_bssn::ADM_BS_phi"),


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:

 ierr += MoLRegisterEvolved(CCTK_VarIndex("ADMBase::alp"),


A matter density $$\rho$$ might not require such a high order scheme and can be evolved using a multi-rate scheme

 ierr += MoLRegisterEvolvedSlow(CCTK_VarIndex("GRHydro::dens"),
CCTK_VarIndex("GRHydro::dendsrhs"));


If maximal or static slicing is used then the lapse is a constrained variable2 :

 ierr += MoLRegisterConstrained(CCTK_VarIndex("ADMBase::alp"));


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 Save and Restore variable:

 ierr += MoLRegisterSaveAndRestore(CCTK_VarIndex("ADMBase::alp"));


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 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

 call CCTK_VarIndex(lapseindex,"ADMBase::alp")
ierr = ierr + MoLChangeToEvolved(lapseindex, lapserhsindex)


It is not required to tell MoL what the lapse is changing 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 ADM_BSSN_removetrA which enforces various constraints (such as the tracefree conformal extrinsic curvature remaining trace free), 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 SYNC at this point. We also convert from the updated BSSN variables back to the standard ADM variables in ADM_BSSN_StandardVariables, and also update the time derivative of the lapse in ADM_BSSN_LapseChange.

### 4 Time evolution methods provided by MoL

The default method is Iterative Crank-Nicholson. There are many ways of implementing this. The standard "ICN" and "Generic"/"ICN" methods both implement the following, assuming an $$N$$ iteration method:

\begin {eqnarray} \label {CactusBase_MoL_eq:icn} {\bf q}^{(0)} & = & {\bf q}^{n}, \\ {\bf q}^{(i)} & = & {\bf q}^{(0)} + \frac {\Delta t}{2} {\bf L}({\bf q}^{(i-1)}), \quad i = 1,\dots ,N-1, \\ {\bf q}^{(N)} & = & {\bf q}^{(N-1)} + \Delta t {\bf L}({\bf q}^{(N-1)}), \\ {\bf q}^{n+1} & = & {\bf q}^{(N)} \end {eqnarray}

The “averaging” ICN method "ICN-avg" instead calculates intermediate steps before averaging:

\begin {eqnarray} \label {CactusBase_MoL_eq:icn-avg} {\bf q}^{(0)} & = & {\bf q}^{n}, \\ \tilde {{\bf q}}^{(i)} & = & \frac {1}{2}\left ( {\bf q}^{(i)} + {\bf q}^{n} \right ), \quad i = 0,\dots ,N-1 \\ {\bf q}^{(i)} & = & {\bf q}^{(0)} + \Delta t {\bf L}(\tilde {{\bf q}}^{(N-1)}), \\ {\bf q}^{n+1} & = & {\bf q}^{(N)} \end {eqnarray}

The Runge-Kutta methods are those typically used in hydrodynamics by, e.g., Shu and others — see [3] for example. Explicitly the first order method is the Euler method:

\begin {eqnarray} \label {CactusBase_MoL_eq:rk1} {\bf q}^{(0)} & = & {\bf q}^{n}, \\ {\bf q}^{(1)} & = & {\bf q}^{(0)} + \Delta t {\bf L}(\tilde {{\bf q}}^{(0)}), \\ {\bf q}^{n+1} & = & {\bf q}^{(1)}. \end {eqnarray}

The second order method is:

\begin {eqnarray} \label {CactusBase_MoL_eq:rk2} {\bf q}^{(0)} & = & {\bf q}^{n}, \\ {\bf q}^{(1)} & = & {\bf q}^{(0)} + \Delta t {\bf L} ({\bf q}^{(0)}), \\ {\bf q}^{(2)} & = & \frac {1}{2} \left ( {\bf q}^{(0)} + {\bf q}^{(1)} + \Delta t {\bf L} ({\bf q}^{(1)}) \right ), \\ {\bf q}^{n+1} & = & {\bf q}^{(2)}. \end {eqnarray}

The third order method is:

\begin {eqnarray} \label {CactusBase_MoL_eq:rk3} {\bf q}^{(0)} & = & {\bf q}^{n}, \\ {\bf q}^{(1)} & = & {\bf q}^{(0)} + \Delta t {\bf L} ({\bf q}^{(0)}), \\ {\bf q}^{(2)} & = & \frac {1}{4} \left ( 3 {\bf q}^{(0)} + {\bf q}^{(1)} + \Delta t {\bf L} ({\bf q}^{(1)}) \right ), \\ {\bf q}^{(3)} & = & \frac {1}{3} \left ( {\bf q}^{(0)} + 2 {\bf q}^{(2)} + 2 \Delta t {\bf L} ({\bf q}^{(2)}) \right ), \\ {\bf q}^{n+1} & = & {\bf q}^{(3)}. \end {eqnarray}

The fourth order method, which is not strictly TVD, is:

\begin {eqnarray} \label {CactusBase_MoL_eq:rk4} {\bf q}^{(0)} & = & {\bf q}^{n}, \\ {\bf q}^{(1)} & = & {\bf q}^{(0)} + \frac {1}{2} \Delta t {\bf L} ({\bf q}^{(0)}), \\ {\bf q}^{(2)} & = & {\bf q}^{(0)} + \frac {1}{2} \Delta t {\bf L} ({\bf q}^{(1)}), \\ {\bf q}^{(3)} & = & {\bf q}^{(0)} + \Delta t {\bf L} ({\bf q}^{(2)}), \\ {\bf q}^{(4)} & = & \frac {1}{6} \left ( - 2 {\bf q}^{(0)} + 2 {\bf q}^{(1)} + 4 {\bf q}^{(2)} + 2 {\bf q}^{(3)} + \Delta t{\bf L} ({\bf q}^{(3)}) \right ), \\ {\bf q}^{n+1} & = & {\bf q}^{(4)}. \end {eqnarray}

#### 4.1 Multirate methods

A scheme for coupling different parts of a system of equations

\begin {eqnarray} \partial _t \mathbf {g} &=& \mathbf {F}(\mathbf {g},\mathbf {q}) \label {eq:CactusBase_MoL_intmetric}, \\ \partial _t \mathbf {q} &=& \mathbf {G}(\mathbf {g},\mathbf {q}) \label {eq:CactusBase_MoL_intmatter}, \end {eqnarray}

representing eg. spacetime and matter variables, respectively, with different RK integrators is given by multirate RK schemes (e.g. [56]). Here, we pursuit the simple Ansatz of performing one RK2 intermediate RHS evaluation for two RK4 intermediate RHS evaluations. That is, the additional RK4 intermediate RHS evaluations simple use the results from the last intermediate RK2 step.

To be more explicit, given the equation \begin {equation} \partial _t y = f(t,y)\,, \end {equation} where $$f$$ corresponds to the RHS possibly including spatial derivatives, we write a generic RK scheme according to

\begin {eqnarray} y_{n+1} &=& y_n + \Delta t \sum _{i=1}^s b_i\, k_i\,, \\ k_i &=& f(t_n + c_i \Delta t\,, y_n + \Delta t \sum _{j=1}^s a_{ij} k_j)\,. \end {eqnarray}

The coefficients $$b_i$$, $$c_i$$, and $$a_{ij}$$ can be written in the standard Butcher notation.

In our multirate scheme, we use two different sets of coefficients. One set of coefficients determines the RK4 scheme used for integrating the spacetime variables (??), the other set determines the RK2 scheme for the hydro variables (??). The coefficients for the RK2 scheme are arranged such that RHS evaluations coincide with RK4 RHS evaluations. We list the corresponding multirate Butcher tableau in Table 1.

Table 1: Butcher tableau for an explicit multirate RK4/RK2 scheme. The right table (separated by the double vertical line) shows the coefficients $$b_i$$ (bottom line), $$c_i$$ (first vertical column), and $$a_{ij}$$ for the classical RK4 scheme. The left table shows the corresponding RK2 coefficients evaluated at timesteps that coincide with RK4 timesteps.
 0 0 0 0 1/2 1/2 0 0 0 1/2 0 1/2 1 1 0 0 1 0 0 1/2 1/2 0 0 1/2 1/3 1/6 1/6 1/3

### 5 Functions provided by MoL

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.

These are all aliased functions. You can get the functions directly through header files, but this feature may be phased out. Using function aliasing is the recommended method.

### MoLRegisterEvolved

Tells MoL that the given GF is in the evolved category with the associated update GF.

Synopsis

C

CCTK_INT ierr = MoLRegisterEvolved(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)
CCTK_INT ierr = MoLRegisterEvolvedSlow(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)


Fortran

CCTK_INT ierr = MoLRegisterEvolved(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)
CCTK_INT ierr = MoLRegisterEvolvedSlow(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

EvolvedIndex Index of the GF to be evolved.
RHSIndex Index of the associated update GF.

Discussion

Should be called in a function scheduled in MoL_Register. Use the Slow variant to register the slow sector of a multirate scheme.

CCTK_VarIndex() Get the variable index.
MoLRegisterSaveAndRestore() Register Save and Restore variables.
MoLRegisterConstrained() Register constrained variables.
MoLChangeToEvolved() Change a variable at runtime to be evolved.
MoLChangeToEvolvedSlow() Change a variable at runtime to be evolved in the slow sector.

Examples

C

ierr = MoLRegisterEvolved(CCTK_VarIndex("wavetoymol::phi"),
CCTK_VarIndex("wavetoymol::phirhs"));


Fortran

call CCTK_VarIndex(EvolvedIndex, "wavetoymol::phi")
call CCTK_VarIndex(RHSIndex, "wavetoymol::phirhs")
ierr = MoLRegisterEvolved(EvolvedIndex, RHSIndex)


### MoLRegisterConstrained

Tells MoL that the given GF is in the constrained category.

Synopsis

C

CCTK_INT ierr = MoLRegisterConstrained(CCTK_INT ConstrainedIndex)


Fortran

CCTK_INT ierr = MoLRegisterConstrained(CCTK_INT ConstrainedIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

ConstrainedIndex Index of the constrained GF.

Discussion

Should be called in a function scheduled in MoL_Register.

CCTK_VarIndex() Get the variable index.
MoLRegisterEvolved() Register evolved variables.
MoLRegisterSaveAndRestore() Register Save and Restore variables.
MoLChangeToConstrained() Change a variable at runtime to be constrained.

Examples

C

ierr = MoLRegisterConstrained(CCTK_VarIndex("ADMBase::alp"));


Fortran

call CCTK_VarIndex(ConstrainedIndex, "ADMBase::alp")
ierr = MoLRegisterConstrained(ConstrainedIndex)


### MoLRegisterSaveAndRestore

Tells MoL that the given GF is in the Save and Restore category.

Synopsis

C

CCTK_INT ierr = MoLRegisterSaveAndRestore(CCTK_INT SandRIndex)


Fortran

CCTK_INT ierr = MoLRegisterSaveAndRestore(CCTK_INT SandRIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

SandRIndex Index of the Save and Restore GF.

Discussion

Should be called in a function scheduled in MoL_Register.

CCTK_VarIndex() Get the variable index.
MoLRegisterEvolved() Register evolved variables.
MoLRegisterConstrained() Register constrained variables.
MoLChangeToSaveAndRestore() Change a variable at runtime to be Save and Restore.

Examples

C

ierr = MoLRegisterSaveAndRestore(CCTK_VarIndex("ADMBase::alp"));


Fortran

call CCTK_VarIndex(SandRIndex, "ADMBase::alp")
ierr = MoLRegisterSaveAndRestore(SandRIndex)


### MoLRegisterEvolvedGroup

Tells MoL that the given group is in the evolved category with the associated update group.

Synopsis

C

CCTK_INT ierr = MoLRegisterEvolvedGroup(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)
CCTK_INT ierr = MoLRegisterEvolvedGroupSlow(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)


Fortran

CCTK_INT ierr = MoLRegisterEvolvedGroup(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)
CCTK_INT ierr = MoLRegisterEvolvedGroupSlow(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

EvolvedIndex Index of the group to be evolved.
RHSIndex Index of the associated update group.

Discussion

Should be called in a function scheduled in MoL_Register. Use the Slow variant to register the slow sector of a multirate scheme.

CCTK_GroupIndex() Get the group index.
MoLRegisterSaveAndRestoreGroup() Register Save and Restore variables.
MoLRegisterConstrainedGroup() Register constrained variables.

Examples

C

ierr = MoLRegisterEvolvedGroup(CCTK_GroupIndex("wavetoymol::scalarevolvemol"),
CCTK_GroupIndex("wavetoymol::scalarevolvemolrhs"));


Fortran

call CCTK_GroupIndex(EvolvedIndex, "wavetoymol::scalarevolvemol")
call CCTK_GroupIndex(RHSIndex, "wavetoymol::scalarevolvemolrhs")
ierr = MoLRegisterEvolvedGroup(EvolvedIndex, RHSIndex)


### MoLRegisterConstrainedGroup

Tells MoL that the given group is in the constrained category.

Synopsis

C

CCTK_INT ierr = MoLRegisterConstrainedGroup(CCTK_INT ConstrainedIndex)


Fortran

CCTK_INT ierr = MoLRegisterConstrainedGroup(CCTK_INT ConstrainedIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

ConstrainedIndex Index of the constrained group.

Discussion

Should be called in a function scheduled in MoL_Register.

CCTK_GroupIndex() Get the group index.
MoLRegisterEvolvedGroup() Register evolved variables.
MoLRegisterSaveAndRestoreGroup() Register Save and Restore variables.
MoLChangeToConstrained() Change a variable at runtime to be constrained.

Examples

C

ierr = MoLRegisterConstrainedGroup(CCTK_GroupIndex("ADMBase::lapse"));


Fortran

call CCTK_GroupIndex(ConstrainedIndex, "ADMBase::lapse")
ierr = MoLRegisterConstrainedGroup(ConstrainedIndex)


### MoLRegisterSaveAndRestoreGroup

Tells MoL that the given group is in the Save and Restore category.

Synopsis

C

CCTK_INT ierr = MoLRegisterSaveAndRestoreGroup(CCTK_INT SandRIndex)


Fortran

CCTK_INT ierr = MoLRegisterSaveAndRestoreGroup(CCTK_INT SandRIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

SandRIndex Index of the save and restore group.

Discussion

Should be called in a function scheduled in MoL_Register.

CCTK_GroupIndex() Get the group index.
MoLRegisterEvolvedGroup() Register evolved variables.
MoLRegisterConstrainedGroup() Register constrained variables.

Examples

C

ierr = MoLRegisterSaveAndRestoreGroup(CCTK_GroupIndex("ADMBase::shift"));


Fortran

call CCTK_GroupIndex(SandRIndex, "ADMBase::shift")
ierr = MoLRegisterSaveAndRestoreGroup(SandRIndex)


### MoLChangeToEvolved

Sets a GF to belong to the evolved category, with the associated update GF. Not used for the initial setting.

Synopsis

C

CCTK_INT ierr = MoLChangeToEvolved(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)
CCTK_INT ierr = MoLChangeToEvolvedSlow(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)


Fortran

CCTK_INT ierr = MoLChangeToEvolvedSlow(CCTK_INT EvolvedIndex,
CCTK_INT RHSIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

EvolvedIndex Index of the evolved GF.
RHSIndex Index of the associated update GF.

Discussion

Should be called in a function scheduled in MoL_PreStep. Note that this function was designed to allow mixed slicings for thorn ADMBase. This set of functions is largely untested and should be used with great care.

CCTK_VarIndex() Get the variable index.
MoLRegisterEvolved() Register evolved variables.
MoLChangeToSaveAndRestore() Change a variable at runtime to be Save and Restore.
MoLChangeToConstrained() Change a variable at runtime to be constrained.

Examples

C

ierr = MoLChangeToEvolved(CCTK_VarIndex("ADMBase::alp"),


Fortran

call CCTK_VarIndex(EvolvedIndex, "ADMBase::alp")
ierr = MoLChangeToEvolved(EvolvedIndex, RHSIndex)


### MoLChangeToConstrained

Sets a GF to belong to the constrained category. Not used for the initial setting.

Synopsis

C

CCTK_INT ierr = MoLChangeToConstrained(CCTK_INT EvolvedIndex)


Fortran

CCTK_INT ierr = MoLChangeToConstrained(CCTK_INT EvolvedIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

ConstrainedIndex Index of the constrained GF.

Discussion

Should be called in a function scheduled in MoL_PreStep. Note that this function was designed to allow mixed slicings for thorn ADMBase. This set of functions is largely untested and should be used with great care.

CCTK_VarIndex() Get the variable index.
MoLRegisterConstrained() Register constrained variables.
MoLChangeToSaveAndRestore() Change a variable at runtime to be Save and Restore.
MoLChangeToEvolved() Change a variable at runtime to be evolved.

Examples

C

ierr = MoLChangeToConstrained(CCTK_VarIndex("ADMBase::alp"));


Fortran

call CCTK_VarIndex(EvolvedIndex, "ADMBase::alp")
ierr = MoLChangeToConstrained(EvolvedIndex)


### MoLChangeToSaveAndRestore

Sets a GF to belong to the Save and Restore category. Not used for the initial setting.

Synopsis

C

CCTK_INT ierr = MoLChangeToSaveAndRestore(CCTK_INT SandRIndex)


Fortran

CCTK_INT ierr = MoLChangeToSaveAndRestore(CCTK_INT SandRIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

SandRIndex Index of the Save and Restore GF.

Discussion

Should be called in a function scheduled in MoL_PreStep. Note that this function was designed to allow mixed slicings for thorn ADMBase. This set of functions is largely untested and should be used with great care.

CCTK_VarIndex() Get the variable index.
MoLRegisterSaveAndRestore() Register Save and Restore variables.
MoLChangeToEvolved() Change a variable at runtime to be evolved.
MoLChangeToConstrained() Change a variable at runtime to be constrained.

Examples

C

ierr = MoLChangeToSaveAndRestore(CCTK_VarIndex("ADMBase::alp"));


Fortran

call CCTK_VarIndex(SandRIndex, "ADMBase::alp")
ierr = MoLChangeToSaveAndRestore(SandRIndex)


### MoLChangeToNone

Sets a GF to belong to the “unknown” category. Not used for the initial setting.

Synopsis

C

CCTK_INT ierr = MoLChangeToNone(CCTK_INT RemoveIndex)


Fortran

CCTK_INT ierr = MoLChangeToNone(CCTK_INT RemoveIndex)


Result

Currently if there is an error, MoL will issue a level 0 warning. No sensible return codes exist. 0 success

Parameters

RemoveIndex Index of the “unknown” GF.

Discussion

Should be called in a function scheduled in MoL_PreStep. Note that this function was designed to allow mixed slicings for thorn ADMBase. This set of functions is largely untested and should be used with great care.

CCTK_VarIndex() Get the variable index.
MoLChangeToEvolved() Change a variable at runtime to be evolved.
MoLChangeToSaveAndRestore() Change a variable at runtime to be Save and Restore.
MoLChangeToConstrained() Change a variable at runtime to be constrained.

Examples

C

ierr = MoLChangeToNone(CCTK_VarIndex("ADMBase::alp"));


Fortran

call CCTK_VarIndex(RemoveIndex, "ADMBase::alp")
ierr = MoLChangeToNone(RemoveIndex)


### MoLQueryEvolvedRHS

Queries MoL for the index of the update variable for given GF in the evolved category.

Synopsis

C

CCTK_INT RHSindex = MoLQueryEvolvedRHS(CCTK_INT EvolvedIndex)


Fortran

CCTK_INT RHSindex = MoLQueryEvolvedRHS(CCTK_INT EvolvedIndex)


Result

If the grid function passed does not exists, MoL will issue a level 0 warning. If the grid function is not of an evolved type (fast or slow sector) $$-1$$ will be returned. Otherwise the variable index of the update GF is returned. $$> 0$$ variable index of update GF

Parameters

EvolvedIndex Index of the GF whose update GF is to be returned.

Discussion

Both slow and fast evolved variables can be queried.

CCTK_VarIndex() Get the variable index.
MoLRegisterEvolved() Register evolved variables.
MoLChangeToEvolvedSlow() Change a variable at runtime to be evolved in the slow sector.

Examples

C

rhsindex = MoLQueryEvolvedRHS(CCTK_VarIndex("wavetoymol::phi"));


Fortran

call CCTK_VarIndex(EvolvedIndex, "wavetoymol::phi")
rhsindex = MoLQueryEvolvedRHS(EvolvedIndex)


### References

[1]   J. Thornburg. Numerical Relativity in Black Hole Spacetimes. Unpublished thesis, University of British Columbia. 1993. Available from http://www.aei.mpg.de/~jthorn/phd/html/phd.html.

[2]   J. Thornburg. A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes – II: Time Evolution. Preprint gr-qc/9906022, submitted to Phys. Rev. D.

[3]   C. Shu. High Order ENO and WENO Schemes for Computational Fluid Dynamics. In T. J. Barth and H. Deconinck, editors High-Order Methods for Computational Physics. Springer, 1999. A related online version can be found under Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws at http://www.icase.edu/library/reports/rdp/97/97-65RDP.tex.refer.html.

[4]   D. W. Neilsen and M. W. Choptuik. Ultrarelativistic fluid dynamics. Class. Quantum Grav., 17: 733–759, 2000.

[5]   M. Schlegel, O. Knoth, M. Arnold, and R. Wolke Journal of Computational and Applied Mathematics, 226, 345, 2009.

[6]   E. Constantinescu and A. Sandu SIAM J. Sci. Comput., 33, 239, 2007.

### 6 Parameters

 ab_initially_reduce_order Scope: private BOOLEAN Description: Reduce order of accuracy initially so that no past timelevels of initial data are required Default: yes

 ab_type Scope: private KEYWORD Description: If using the the AB method, which sort Range Default: 1 1 same as forward Euler 2 second order 3 third order 4 fourth order 5 fifth order

 adaptive_stepsize Scope: private BOOLEAN Description: Choose the time step size adaptively Default: no

 copy_id_after_mol_poststep Scope: private BOOLEAN Description: if initial_data_is_crap is true, *when* should we copy the current time level to all previous time levels: false ==> copy *before* MoL_PostStep (default, matches old behavior) true ==> copy *after* MoL_PostStep (maybe preferable for new code) Default: no

 disable_prolongation Scope: private BOOLEAN Description: If Mesh refinement is enabled should we use buffer zones in intermediate steps? Default: yes

 generic_method_descriptor Scope: private STRING Description: A string used to create a table containing the description of the generic method Range Default: GenericIntermediateSteps = 2 GenericAlphaCoeffs = { 1.0 0.0 0.5 0.5 } GenericBetaCoeffs = { 1.0 0.5 } .* Should contain the Alpha and Beta arrays, and the number of intermediate steps

 generic_type Scope: private KEYWORD Description: If using the generic method, which sort Range Default: RK RK One of the standard TVD Runge-Kutta methods ICN Iterative Crank Nicholson as a generic method Table Given from the generic method descriptor parameter Classic RK3 Efficient RK3 - classical version

 icn_avg_swapped Scope: private BOOLEAN Description: Use swapped averages in ICN method? Default: no

 icn_avg_theta Scope: private REAL Description: theta of averaged ICN method, usually 0.5 Range Default: 0.5 0:1 0 <= theta <= 1

 init_rhs_zero Scope: private BOOLEAN Description: Initialise the RHS to zero Default: yes

 initial_data_is_crap Scope: private BOOLEAN Description: If the initial data routine fails to set up the previous time levels, copy the current backwards Default: no

 maximum_absolute_error Scope: private REAL Description: Maximum allowed absolute error for adaptive stepsize control Range Default: 1.0e-6 0.0:*)

 maximum_decrease Scope: private REAL Description: Maximum stepsize decrease factor Range Default: 10.0 (1.0:*) should be larger than one

 maximum_increase Scope: private REAL Description: Maximum stepsize increase factor Range Default: 5.0 (1.0:*) should be larger than one

 maximum_relative_error Scope: private REAL Description: Maximum allowed relative error for adaptive stepsize control Range Default: 1.0e-6 0.0:*)

 mol_intermediate_steps Scope: private INT Description: Number of intermediate steps taken by the ODE method Range Default: 3 1:* Anything greater than 1

 mol_memory_always_on Scope: private BOOLEAN Description: Do we keep the scratch arrays allocated all the time? Default: yes

 mol_nan_check Scope: private BOOLEAN Description: Should the RHS GFs be checked for NaNs? Default: no

 mol_tiny Scope: private REAL Description: Effective local machine zero; required by generic solvers Range Default: 1.e-15 0:* Defaults to 1.e-15

 ode_method Scope: private KEYWORD Description: The ODE method use by MoL to do time integration Range Default: ICN Generic Generic Shu-Osher Runge-Kutta type ICN Iterative Crank Nicholson ICN-avg Iterative Crank Nicholson with averaging Euler Euler RK2 Efficient RK2 RK2-central Central RK2 RK3 Efficient RK3 RK4 Efficient RK4 RK45 RK45 (Fehlberg) with error estimation RK45CK RK45CK (Cash-Karp) with error estimation RK65 RK65 with error estimation RK87 RK87 with error estimation AB Adams-Bashforth RK2-MR-2:1 2nd order 2:1 multirate RK scheme based on RK2 due to Schlegel et al 2009. This requires init_RHS_zero=’no’. RK4-MR-2:1 3rd order 2:1 multirate RK scheme based on RK43 due to Schlegel et al 2009. This requires init_RHS_zero=’no’. RK4-RK2 RK4 as fast method and RK2 as slow method

 rhs_error_weight Scope: private REAL Description: Weight of the RHS in the relative error calculation Range Default: 1.0 0.0:* should be between zero and one

 run_mol_poststep_in_post_recover_variables Scope: private BOOLEAN Description: Schedule the PostStep parts after recovery so that symmetries are automatically done correctly. Default: yes

 safety_factor Scope: private REAL Description: Safety factor for stepsize control Range Default: 0.9 (0.0:*) should be less than one

 set_id_boundaries Scope: private BOOLEAN Description: Should boundaries be overwritten (via synchronization, prolongation, boundary conditions) by MoL? Default: yes

 skip_initial_copy Scope: private BOOLEAN Description: Skip initial copy from previous to current time level Default: no

 verbose Scope: private KEYWORD Description: How verbose should MoL be? Range Default: normal none No output at all (not implemented) normal Standard verbosity register List the variables registered as well extreme Everything you never wanted to know

 mol_max_evolved_array_size Scope: restricted INT Description: The maximum total size of any grid arrays to be evolved Range Default: (none) 0:* Anything non negative. Accumulated by other thorns

 mol_num_arrayconstrained_vars Scope: restricted INT Description: The maximum number of array constrained variables with timelevels that MoL needs to know about (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_arrayevolved_vars Scope: restricted INT Description: The maximum number of array variables to be evolved by MoL (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_arraysaveandrestore_vars Scope: restricted INT Description: The maximum number of array variables to be evolved outside of MoL but that MoL needs to know about (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_constrained_vars Scope: restricted INT Description: The maximum number of constrained variables with timelevels that MoL needs to know about (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_evolved_vars Scope: restricted INT Description: The maximum number of variables to be evolved by MoL (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_evolved_vars_slow Scope: restricted INT Description: The maximum number of ’slow’ variables to be evolved by MoL (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_saveandrestore_vars Scope: restricted INT Description: The maximum number of variables to be evolved outside of MoL but that MoL needs to know about (DPRECATED) Range Default: (none) 0:* Anything non negative. Added to by other thorns.

 mol_num_scratch_levels Scope: restricted INT Description: Number of scratch levels required by the ODE method Range Default: (none) 0:* Anything non negative

 cctk_initial_time Scope: shared from CACTUS REAL

 presync_mode Scope: shared from CACTUS KEYWORD

### 7 Interfaces

Implements:

methodoflines

#### Grid Variables

##### 7.0.1 PRIVATE GROUPS
 Group Names Variable Names Details rkalphacoefficients RKAlphaCoefficients compact 0 dimensions 2 distribution CONSTANT group type ARRAY size MOL_INTERMEDIATE_STEPS size MOL_NUM_SCRATCH_LEVELS+1 tags Checkpoint=”no” timelevels 1 variable type REAL rkbetacoefficients RKBetaCoefficients compact 0 dimensions 1 distribution CONSTANT group type ARRAY size MOL_INTERMEDIATE_STEPS tags Checkpoint=”no” timelevels 1 variable type REAL mol_counters compact 0 MoL_Intermediate_Step description The counter for the time integration method MoL_Stepsize_Bad dimensions 0 MoL_SlowStep distribution CONSTANT MoL_SlowPostStep group type SCALAR tags Checkpoint=”no” timelevels 1 variable type INT mol_original_time compact 0 Original_Time description The original time and delta time which are reset by MoL during evolution Original_Delta_Time dimensions 0 distribution CONSTANT group type SCALAR tags Checkpoint=”no” timelevels 1 variable type REAL scratchspaceslow ScratchSpaceSlow compact 0 dimensions 3 distribution DEFAULT group type GF tags Prolongation=”None” Checkpoint=”no” timelevels 99 vararray_size MoL_Num_Scratch_Levels variable type REAL scratchspace ScratchSpace compact 0 dimensions 3 distribution DEFAULT group type GF tags Prolongation=”None” Checkpoint=”no” timelevels 99 vararray_size MoL_Num_Scratch_Levels variable type REAL
 Group Names Variable Names Details sandrscratchspace SandRScratchSpace compact 0 dimensions 3 distribution DEFAULT group type GF tags Prolongation=”None” Checkpoint=”no” timelevels 99 variable type REAL errorestimate ErrorEstimate compact 0 dimensions 3 distribution DEFAULT group type GF tags Prolongation=”None” Checkpoint=”no” timelevels 99 variable type REAL errorscalars compact 0 Error description Global error estimate Count dimensions 0 EstimatedDt distribution CONSTANT group type SCALAR timelevels 1 variable type REAL

Provides:

MoLRegisterEvolved to

MoLRegisterEvolvedSlow to

MoLRegisterConstrained to

MoLRegisterSaveAndRestore to

MoLRegisterEvolvedGroup to

MoLRegisterEvolvedGroupSlow to

MoLRegisterConstrainedGroup to

MoLRegisterSaveAndRestoreGroup to

MoLChangeToEvolved to

MoLChangeToEvolvedSlow to

MoLChangeToConstrained to

MoLChangeToSaveAndRestore to

MoLChangeToNone to

MoLQueryEvolvedRHS to

MoLNumIntegratorSubsteps to

### 8 Schedule

This section lists all the variables which are assigned storage by thorn CactusNumerical/MoL. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

#### Storage

 Always: Conditional: MoL_Counters MoL_Original_Time RKAlphaCoefficients RKBetaCoefficients ErrorScalars ErrorScalars ErrorScalars

#### Scheduled Functions

CCTK_STARTUP (conditional)

mol_startup

startup banner

 Language: c Type: function

CCTK_PARAMCHECK (conditional)

mol_paramcheck

basic parameter checking

 Language: c Type: function

CCTK_INITIAL (conditional)

mol_startloop

initialise the step size control

 Language: c Options: level Type: function Writes: mol_stepsize_bad estimateddt

CCTK_EVOL (conditional)

mol_evolution

a single cactus evolution step using mol

MoL_Evolution (conditional)

mol_startstep

mol internal setup for the evolution step

 Type: group

MoL_StartStep (conditional)

mol_setcounter

set the counter for the ode method to loop over

 Language: c Options: level Type: function Writes: mol_intermediate_step mol_slowstep mol_slowpoststep

MoL_StartStep (conditional)

mol_settime

ensure the correct time and timestep are used

 Language: c Options: level Reads: rkalphacoefficients rkbetacoefficients mol_intermediate_step Type: function Writes: mol_original_time

MoL_StartStep (conditional)

mol_allocatescratchspace

allocate storage for scratch levels

 Language: c Options: level Type: function

MoL_Evolution (conditional)

mol_prestep

physics thorns can schedule preloop setup routines in here

 After: mol_startstep Before: mol_step Type: group

MoL_Evolution (conditional)

mol_allocatescratch

allocate sufficient space for array scratch variables

 After: mol_prestep Before: mol_step Language: c Type: function

MoL_Evolution (conditional)

mol_initialcopy

ensure the data is in the correct timelevel

 After: mol_prestep mol_allocatescratch Before: mol_step Language: c Type: function

MoL_Evolution (conditional)

mol_updatevalidforinitialcopy

automatically update valid regions based on mol

 After: mol_initialcopy Before: mol_step Language: c Type: function

CCTK_WRAGH (conditional)

mol_setupindexarrays

set up the mol bookkeeping index arrays

 Language: c Type: function Writes: mol_original_time mol_slowpoststep mol_slowstep

MoL_Evolution (conditional)

mol_step

the loop over the intermediate steps for the ode integrator

 After: mol_prestep Type: group While: mol::mol_intermediate_step

MoL_Step (conditional)

mol_icnaverage

averages the time levels for the averaging icn method

 Before: mol_calcrhs Language: c Reads: mol_intermediate_step Type: function

MoL_Step (conditional)

mol_initrhs

initialise the rhs functions

 Before: mol_calcrhs Language: c Type: function

MoL_Step (conditional)

mol_calcrhs

physics thorns schedule the calculation of the discrete spatial operator in here

 Type: group

MoL_Step (conditional)

mol_postrhs

modify rhs functions

 After: mol_calcrhs Before: mol_nancheck mol_add Type: group

MoL_Step (conditional)

mol_rhsboundaries

any ’final’ modifications to the rhs functions (boundaries etc.)

 After: mol_postrhs Before: mol_nancheck mol_add Type: group

MoL_Step (conditional)

mol_nancheck

check the rhs gfs for nans

MoL_Step (conditional)

updates calculated with a generic method

 After: mol_calcrhs Before: mol_poststep mol_poststepmodify Language: c Reads: mol_intermediate_step original_delta_time rkalphacoefficients rkbetacoefficients Type: function

MoL_Step (conditional)

updates calculated with the euler method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step Type: function

MoL_Step (conditional)

updates calculated with the efficient runge-kutta 2 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step Type: function

CCTK_WRAGH (conditional)

mol_setuprkcoefficients

initialize the generic runge-kutta coefficients

 Language: c Options: global Storage: rkalphacoefficients rkbetacoefficients Type: function Writes: rkalphacoefficients rkbetacoefficients

MoL_Step (conditional)

updates calculated with the central runge-kutta 2 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: original_delta_time mol_intermediate_step Type: function

MoL_Step (conditional)

updates calculated with the efficient runge-kutta 3 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step Type: function

MoL_Step (conditional)

updates calculated with the efficient runge-kutta 4 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

MoL_Step (conditional)

updates calculated with the runge-kutta 45 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

MoL_Step (conditional)

updates calculated with the runge-kutta 65 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

MoL_Step (conditional)

updates calculated with the runge-kutta 87 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

MoL_Step (conditional)

updates calculated with the efficient icn method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step Type: function

MoL_Step (conditional)

updates calculated with the averaging icn method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step Type: function

MoL_Step (conditional)

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step Type: function

MoL_Step (conditional)

updates calculated with the multirate runge-kutta 2 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

CCTK_WRAGH (conditional)

mol_setschedulestatus

set the flag so it is ok to register with mol

 After: mol_setupindexarrays Language: c Options: global Type: function

MoL_Step (conditional)

updates calculated with the multirate runge-kutta 4 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

MoL_Step (conditional)

updates calculated with the multirate rk4/rk2 method

 After: mol_calcrhs Before: mol_poststep Language: c Reads: mol_intermediate_step original_delta_time Type: function

MoL_Step (conditional)

automatically update valid regions based on mol

 After: mol_add Language: c Type: function

MoL_Step (conditional)

mol_poststep

the group for physics thorns to schedule boundary calls etc.

CCTK_POSTINITIAL (conditional)

mol_poststep

ensure that everything is correct after the initial data have been set up

 Type: group

CCTK_POSTREGRID (conditional)

mol_poststep

ensure that everything is correct after regridding

 Type: group

CCTK_POSTRESTRICTINITIAL (conditional)

mol_poststep

ensure that everything is correct after restriction

 Type: group

CCTK_POSTRESTRICT (conditional)

mol_poststep

ensure that everything is correct after restriction

 Type: group

CCTK_POST_RECOVER_VARIABLES (conditional)

mol_poststep

ensure that everything is correct after recovery

 Type: group

MoL_Step

mol_poststepmodify

the group for physics thorns to schedule enforcing constraints

 After: mol_add Before: mol_poststep Type: group

CCTK_WRAGH (conditional)

mol_register

the group where physics thorns register variables with mol

 After: mol_setschedulestatus Type: group

CCTK_POSTINITIAL

mol_poststepmodify

the group for physics thorns to schedule enforcing constraints

 Before: mol_poststep Type: group

CCTK_POSTINITIAL

mol_pseudoevolution

calculate pseudo-evolved quantities

 After: mol_poststep Type: group

CCTK_EVOL

mol_pseudoevolution

calculate pseudo-evolved quantities

 After: mol_evolution Type: group

CCTK_POSTREGRIDINITIAL

mol_pseudoevolutionboundaries

apply boundary conditions to pseudo-evolved quantities

 After: mol_poststep Type: group

CCTK_POSTREGRID

mol_pseudoevolutionboundaries

apply boundary conditions to pseudo-evolved quantities

 After: mol_poststep Type: group

CCTK_POSTRESTRICTINITIAL

mol_pseudoevolutionboundaries

apply boundary conditions to pseudo-evolved quantities

 After: mol_poststep Type: group

CCTK_POSTRESTRICT

mol_pseudoevolutionboundaries

apply boundary conditions to pseudo-evolved quantities

 After: mol_poststep Type: group

MoL_Step

mol_decrementcounter

alter the counter number

 After: mol_add Before: mol_poststep mol_poststepmodify Language: c Options: level Type: function Writes: mol_intermediate_step mol_slowstep mol_slowpoststep

MoL_Step

mol_resettime

if necessary, change the time

 After: mol_decrementcounter Before: mol_poststep mol_poststepmodify Language: c Options: level Reads: mol_original_time rkalphacoefficients rkbetacoefficients mol_intermediate_step Type: function

MoL_Step

mol_resetdeltatime

if necessary, change the timestep

 After: mol_poststep mol_poststepmodify Language: c Options: level Reads: original_delta_time rkalphacoefficients rkbetacoefficients mol_intermediate_step Type: function

CCTK_WRAGH (conditional)

mol_reportnumbervariables

report how many of each type of variable there are

 After: mol_register Language: c Options: meta Type: function

MoL_Evolution

mol_restoresandr

restoring the save and restore variables to the original state

 After: mol_reduceadaptiveerror mol_finishloop Language: c Type: function

MoL_Evolution

mol_freescratchspace

free storage for scratch levels

 After: mol_restoresandr Language: c Options: level Type: function

MoL_Evolution (conditional)

control the step size: initialize error check variables

 After: mol_step Language: c Options: level Type: function Writes: error count

MoL_Evolution (conditional)

control the step size: compute error check variables

MoL_Evolution (conditional)

control the step size: reduce error check variables

CCTK_POSTSTEP (conditional)

mol_setestimateddt

control the step size: set the new timestep

 Language: c Options: level Reads: estimateddt Type: function

MoL_Evolution (conditional)

mol_finishloop

control the step size

 After: mol_step Language: c Options: level Type: function Writes: mol_stepsize_bad mol_slowpoststep mol_slowstep

CCTK_TERMINATE

mol_freeindexarrays

free the mol bookkeeping index arrays

 Before: driver_terminate Language: c Type: function

CCTK_POSTINITIAL (conditional)

mol_fillalllevels

a bad routine. fills all previous timelevels with data copied from the current.

 After: mol_poststep Language: c Type: function

CCTK_POSTINITIAL (conditional)

mol_fillalllevels

a bad routine. fills all previous timelevels with data copied from the current.

 Before: mol_poststep Language: c Type: function

CCTK_EVOL (conditional)

mol_startloop

initialise the step size control

 Before: mol_evolution Language: c Options: level Type: function Writes: mol_stepsize_bad estimateddt