Integrator Interface
The integrator interface gives one the ability to interactively step through the numerical solving of a differential equation. Through this interface, one can easily monitor results, modify the problem during a run, and dynamically continue solving as one sees fit.
Initialization and Stepping
To initialize an integrator, use the syntax:
integrator = init(prob,alg;kwargs...)
The keyword args which are accepted are the same as the solver options used by solve
and the returned value is an integrator
which satisfies typeof(integrator)<:DEIntegrator
. One can manually choose to step via the step!
command:
step!(integrator)
which will take one successful step. Additonally:
step!(integrator,dt[,stop_at_tdt=false])
passing a dt
will make the integrator keep stepping until integrator.t+dt
, and setting stop_at_tdt=true
will add a tstop
to force it to step to integrator.t+dt
To check whether or not the integration step was successful, you can call check_error(integrator)
which returns one of the return codes.
This type also implements an iterator interface, so one can step n
times (or to the last tstop
) using the take
iterator:
for i in take(integrator,n) end
One can loop to the end by using solve!(integrator)
or using the iterator interface:
for i in integrator end
In addition, some helper iterators are provided to help monitor the solution. For example, the tuples
iterator lets you view the values:
for (u,t) in tuples(integrator)
@show u,t
end
and the intervals
iterator lets you view the full interval:
for (uprev,tprev,u,t) in intervals(integrator)
@show tprev,t
end
Additionally, you can make the iterator return specific time points via the TimeChoiceIterator
:
ts = range(0, stop=1, length=11)
for (u,t) in TimeChoiceIterator(integrator,ts)
@show u,t
end
Lastly, one can dynamically control the "endpoint". The initialization simply makes prob.tspan[2]
the last value of tstop
, and many of the iterators are made to stop at the final tstop
value. However, step!
will always take a step, and one can dynamically add new values of tstops
by modifiying the variable in the options field: add_tstop!(integrator,new_t)
.
Finally, to solve to the last tstop
, call solve!(integrator)
. Doing init
and then solve!
is equivalent to solve
.
SciMLBase.step!
— Functionstep!(integ::DEIntegrator [, dt [, stop_at_tdt]])
Perform one (successful) step on the integrator.
Alternative, if a dt
is given, then step!
the integrator until there is a temporal difference ≥ dt
in integ.t
. When true
is passed to the optional third argument, the integrator advances exactly dt
.
SciMLBase.check_error
— Functioncheck_error(integrator)
Check state of integrator
and return one of the Return Codes
SciMLBase.check_error!
— Functioncheck_error!(integrator)
Same as check_error
but also set solution's return code (integrator.sol.retcode
) and run postamble!
.
Handing Integrators
The integrator<:DEIntegrator
type holds all of the information for the intermediate solution of the differential equation. Useful fields are:
t
- time of the proposed stepu
- value at the proposed stepp
- user-provided dataopts
- common solver optionsalg
- the algorithm associated with the solutionf
- the function being solvedsol
- the current state of the solutiontprev
- the last timepointuprev
- the value at the last timepointtdir
- the sign for the direction of time
The p
is the data which is provided by the user as a keyword arg in init
. opts
holds all of the common solver options, and can be mutated to change the solver characteristics. For example, to modify the absolute tolerance for the future timesteps, one can do:
integrator.opts.abstol = 1e-9
The sol
field holds the current solution. This current solution includes the interpolation function if available, and thus integrator.sol(t)
lets one interpolate efficiently over the whole current solution. Additionally, a a "current interval interpolation function" is provided on the integrator
type via integrator(t,deriv::Type=Val{0};idxs=nothing,continuity=:left)
. This uses only the solver information from the interval [tprev,t]
to compute the interpolation, and is allowed to extrapolate beyond that interval.
Note about mutating
Be cautious: one should not directly mutate the t
and u
fields of the integrator. Doing so will destroy the accuracy of the interpolator and can harm certain algorithms. Instead if one wants to introduce discontinuous changes, one should use the callbacks. Modifications within a callback affect!
surrounded by saves provides an error-free handling of the discontinuity.
As low-level alternative to the callbacks, one can use set_t!
, set_u!
and set_ut!
to mutate integrator states. Note that certain integrators may not have efficient ways to modify u
and t
. In such case, set_*!
are as inefficient as reinit!
.
SciMLBase.set_t!
— Functionset_t!(integrator::DEIntegrator, t)
Set current time point of the integrator
to t
.
SciMLBase.set_u!
— Functionset_u!(integrator::DEIntegrator, u)
Set current state of the integrator
to u
.
SciMLBase.set_ut!
— Functionset_ut!(integrator::DEIntegrator, u, t)
Set current state of the integrator
to u
and t
Integrator vs Solution
The integrator and the solution have very different actions because they have very different meanings. The typeof(sol) <: DESolution
type is a type with history: it stores all of the (requested) timepoints and interpolates/acts using the values closest in time. On the other hand, the typeof(integrator)<:DEIntegrator
type is a local object. It only knows the times of the interval it currently spans, the current caches and values, and the current state of the solver (the current options, tolerances, etc.). These serve very different purposes:
- The
integrator
's interpolation can extrapolate, both forward and backward in in time. This is used to estimate events and is internally used for predictions. - The
integrator
is fully mutable upon iteration. This means that every time an iterator affect is used, it will take timesteps from the current time. This means thatfirst(integrator)!=first(integrator)
since theintegrator
will step once to evaluate the left and then step once more (not backtracking). This allows the iterator to keep dynamically stepping, though one should note that it may violate some immutablity assumptions commonly made about iterators.
If one wants the solution object, then one can find it in integrator.sol
.
Function Interface
In addition to the type interface, a function interface is provided which allows for safe modifications of the integrator type, and allows for uniform usage throughout the ecosystem (for packages/algorithms which implement the functions). The following functions make up the interface:
Saving Controls
SciMLBase.savevalues!
— Functionsavevalues!(integrator::DEIntegrator,
force_save=false) -> Tuple{Bool, Bool}
Try to save the state and time variables at the current time point, or the saveat
point by using interpolation when appropriate. It returns a tuple that is (saved, savedexactly)
. If savevalues!
saved value, then saved
is true, and if savevalues!
saved at the current time point, then savedexactly
is true.
The saving priority/order is as follows:
save_on
saveat
force_save
save_everystep
Caches
SciMLBase.get_tmp_cache
— Functionget_tmp_cache(i::DEIntegrator)
Returns a tuple of internal cache vectors which are safe to use as temporary arrays. This should be used for integrator interface and callbacks which need arrays to write into in order to be non-allocating. The length of the tuple is dependent on the method.
SciMLBase.full_cache
— Functionfull_cache(i::DEIntegrator)
Returns an iterator over the cache arrays of the method. This can be used to change internal values as needed.
Stepping Controls
SciMLBase.u_modified!
— Functionu_modified!(i::DEIntegrator,bool)
Sets bool
which states whether a change to u
occurred, allowing the solver to handle the discontinuity. By default, this is assumed to be true if a callback is used. This will result in the re-calculation of the derivative at t+dt
, which is not necessary if the algorithm is FSAL and u
does not experience a discontinuous change at the end of the interval. Thus if u
is unmodified in a callback, a single call to the derivative calculation can be eliminated by u_modified!(integrator,false)
.
SciMLBase.get_proposed_dt
— Functionget_proposed_dt(i::DEIntegrator)
Gets the proposed dt
for the next timestep.
SciMLBase.set_proposed_dt!
— Functionset_proposed_dt(i::DEIntegrator,dt)
set_proposed_dt(i::DEIntegrator,i2::DEIntegrator)
Sets the proposed dt
for the next timestep. If second argument isa DEIntegrator
then it sets the timestepping of first argument to match that of second one. Note that due to PI control and step acceleration this is more than matching the factors in most cases.
SciMLBase.terminate!
— Functionterminate!(i::DEIntegrator[, retcode = :Terminated])
Terminates the integrator by emptying tstops
. This can be used in events and callbacks to immediately end the solution process. Optionally, retcode
may be specified (see: Return Codes (RetCodes)).
SciMLBase.change_t_via_interpolation!
— Functionchange_t_via_interpolation!(integrator::DEIntegrator,t,modify_save_endpoint=Val{false})
Modifies the current t
and changes all of the corresponding values using the local interpolation. If the current solution has already been saved, one can provide the optional value modify_save_endpoint
to also modify the endpoint of sol
in the same manner.
SciMLBase.add_tstop!
— Functionadd_tstop!(i::DEIntegrator,t)
Adds a tstop
at time t
.
SciMLBase.add_saveat!
— Functionadd_saveat!(i::DEIntegrator,t)
Adds a saveat
time point at t
.
Resizing
Base.resize!
— Functionresize!(integrator::DEIntegrator,k::Int)
Resizes the DE to a size k
. This chops off the end of the array, or adds blank values at the end, depending on whether k > length(integrator.u)
.
Base.deleteat!
— Functiondeleteat!(integrator::DEIntegrator,idxs)
Shrinks the ODE by deleting the idxs
components.
SciMLBase.addat!
— Functionaddat!(integrator::DEIntegrator,idxs,val)
Grows the ODE by adding the idxs
components. Must be contiguous indices.
SciMLBase.resize_non_user_cache!
— Functionresize_non_user_cache!(integrator::DEIntegrator,k::Int)
Resizes the non-user facing caches to be compatible with a DE of size k
. This includes resizing Jacobian caches.
In many cases, resize!
simply resizes full_cache
variables and then calls this function. This finer control is required for some AbstractArray
operations.
SciMLBase.deleteat_non_user_cache!
— Functiondeleteat_non_user_cache!(integrator::DEIntegrator,idxs)
deleteat!
s the non-user facing caches at indices idxs
. This includes resizing Jacobian caches.
In many cases, deleteat!
simply deleteat!
s full_cache
variables and then calls this function. This finer control is required for some AbstractArray
operations.
SciMLBase.addat_non_user_cache!
— Functionaddat_non_user_cache!(i::DEIntegrator,idxs)
addat!
s the non-user facing caches at indices idxs
. This includes resizing Jacobian caches.
In many cases, addat!
simply addat!
s full_cache
variables and then calls this function. This finer control is required for some AbstractArray
operations.
Reinitialization
SciMLBase.reinit!
— Functionreinit!(integrator::DEIntegrator,args...; kwargs...)
The reinit function lets you restart the integration at a new value.
Arguments
u0
: Value ofu
to start at. Default value isintegrator.sol.prob.u0
Keyword Arguments
t0
: Starting timepoint. Default value isintegrator.sol.prob.tspan[1]
tf
: Ending timepoint. Default value isintegrator.sol.prob.tspan[2]
erase_sol=true
: Whether to start with no other values in the solution, or keep the previous solution.tstops
,d_discontinuities
, &saveat
: Cache where these are stored. Default is the original cache.reset_dt
: Set whether to reset the current value ofdt
using the automaticdt
determination algorithm. Default is(integrator.dtcache == zero(integrator.dt)) && integrator.opts.adaptive
reinit_callbacks
: Set whether to run the callback initializations again (andinitialize_save
is for that). Default istrue
.reinit_cache
: Set whether to re-run the cache initialization function (i.e. resetting FSAL, not allocating vectors) which should usually be true for correctness. Default istrue
.
Additionally, once can access auto_dt_reset!
which will run the auto dt
initialization algorithm.
SciMLBase.auto_dt_reset!
— Functionauto_dt_reset!(integrator::DEIntegrator)
Run the auto dt
initialization algorithm.
Misc
SciMLBase.get_du
— Functionget_du(i::DEIntegrator)
Returns the derivative at t
.
SciMLBase.get_du!
— Functionget_du!(out,i::DEIntegrator)
Write the current derivative at t
into out
.
Note that not all of these functions will be implemented for every algorithm. Some have hard limitations. For example, Sundials.jl cannot resize problems. When a function is not limited, an error will be thrown.
Additional Options
The following options can additionally be specified in init
(or be mutated in the opts
) for further control of the integrator:
advance_to_tstop
: This makesstep!
continue to the next value intstop
.stop_at_next_tstop
: This forces the iterators to stop at the next value oftstop
.
For example, if one wants to iterate but only stop at specific values, one can choose:
integrator = init(prob,Tsit5();dt=1//2^(4),tstops=[0.5],advance_to_tstop=true)
for (u,t) in tuples(integrator)
@test t ∈ [0.5,1.0]
end
which will only enter the loop body at the values in tstops
(here, prob.tspan[2]==1.0
and thus there are two values of tstops
which are hit). Addtionally, one can solve!
only to 0.5
via:
integrator = init(prob,Tsit5();dt=1//2^(4),tstops=[0.5])
integrator.opts.stop_at_next_tstop = true
solve!(integrator)
Plot Recipe
Like the DESolution
type, a plot recipe is provided for the DEIntegrator
type. Since the DEIntegrator
type is a local state type on the current interval, plot(integrator)
returns the solution on the current interval. The same options for the plot recipe are provided as for sol
, meaning one can choose variables via the vars
keyword argument, or change the plotdensity
/ turn on/off denseplot
.
Additionally, since the integrator
is an iterator, this can be used in the Plots.jl animate
command to iteratively build an animation of the solution while solving the differential equation.
For an example of manually chaining together the iterator interface and plotting, one should try the following:
using DifferentialEquations, DiffEqProblemLibrary, Plots
# Linear ODE which starts at 0.5 and solves from t=0.0 to t=1.0
prob = ODEProblem((u,p,t)->1.01u,0.5,(0.0,1.0))
using Plots
integrator = init(prob,Tsit5();dt=1//2^(4),tstops=[0.5])
pyplot(show=true)
plot(integrator)
for i in integrator
display(plot!(integrator,vars=(0,1),legend=false))
end
step!(integrator); plot!(integrator,vars=(0,1),legend=false)
savefig("iteratorplot.png")