integrator.md

[Integrator Interface](@id integrator)

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](@ref 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. Additionally:

step!(integrator, dt, false)

passing a dt will make the integrator continue to step until at least integrator.t+dt, and passing true as the third argument will add a tstop to force it to step up to integrator.t+dt, exactly.

To check whether the integration step was successful, you can call check_error(integrator) which returns one of the [return codes](@ref retcodes).

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 modifying 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!
SciMLBase.check_error
SciMLBase.check_error!

Handling Integrators

The integrator<:DEIntegrator type holds all the information for the intermediate solution of the differential equation. Useful fields are:

• t - time of the proposed step
• u - value at the proposed step
• p - user-provided data
• opts - common solver options
• alg - the algorithm associated with the solution
• f - the function being solved
• sol - the current state of the solution
• tprev - the last timepoint
• uprev - the value at the last timepoint
• tdir - the sign for the direction of time

The function f is usually a wrapper of the function provided when creating the specific problem. For example, when solving an ODEProblem, f will be an ODEFunction. To access the right-hand side function provided by the user when creating the ODEProblem, please use SciMLBase.unwrapped_f(integrator.f.f).

The p is the (parameter) data which is provided by the user as a keyword arg in init. opts holds all 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 “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](@ref 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!
SciMLBase.set_u!
SciMLBase.set_ut!

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 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 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 that first(integrator)!=first(integrator) since the integrator 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 immutability 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

savevalues!

Caches

get_tmp_cache
full_cache

[Stepping Controls](@id stepping_controls)

u_modified!
get_proposed_dt
set_proposed_dt!
terminate!
change_t_via_interpolation!
add_tstop!
has_tstop
first_tstop
pop_tstop!
add_saveat!

Resizing

resize!
deleteat!
addat!
resize_non_user_cache!
deleteat_non_user_cache!
addat_non_user_cache!

Reinitialization

reinit!
auto_dt_reset!

Misc

get_du
get_du!

!!! warning

Note that not all 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 makes step! continue to the next value in tstop.
• stop_at_next_tstop: This forces the iterators to stop at the next value of tstop.

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). Additionally, 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 idxs 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, idxs = (0, 1), legend = false))
end
step!(integrator);
plot!(integrator, idxs = (0, 1), legend = false);
savefig("iteratorplot.png")