Add option to specify maximum number of consecutively rejected steps

[efaulhaber]

Without default dtmin (https://github.com/SciML/OrdinaryDiffEq.jl/pull/2098), automatic time stepping methods reduce the time step below eps() at instabilities and reject all steps infinitely. MWE:

f(u, p, t) = (print("x"); t > 0.1 ? 1e30 : 0.1);
ode = ODEProblem(f, 0.0, (0.0, 2.0));
cb = DiscreteCallback((u, t, integrator) -> true, integrator -> println("\n$(integrator.iter)"), save_positions=(false, false));

julia> solve(ode, Tsit5(), save_everystep=false, maxiters=1000, callback=cb);
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└ @ SciMLBase ~/.julia/packages/SciMLBase/NjslX/src/integrator_interface.jl:600

As shown by the many x at the end, f is still called, but there are no more accepted steps to trigger the callback until we reach maxiters. In simulation frameworks, this will cause apparent freezing of the simulation, as suddenly no more output is generated when an instability is reached, but the simulation doesn't terminate.

I suggest adding an option to limit the maximum number of consecutively rejected steps to terminate time integration in these cases where it gets stuck in an infinite loop until maxiters is reached (which might take a very long time for large simulations and default maxiters).

Note that setting dtmin is not an option, as perfectly stable simulations are terminated when different callbacks happen to add tstops that are very close together (#1616):

julia> solve(ode, Tsit5(), tstops=[0.05, 0.05 + eps()], dtmin=eps())
┌ Warning: dt(2.220446049250313e-16) <= dtmin(2.220446049250313e-16) at t=0.05, and step error estimate = 1.6870185803874452e-14. Aborting. There is either an error in your model specification or the true solution is unstable.
└ @ SciMLBase ~/.julia/packages/SciMLBase/NjslX/src/integrator_interface.jl:619
retcode: DtLessThanMin
Interpolation: specialized 4th order "free" interpolation
t: 5-element Vector{Float64}:
 0.0
 9.999999999999999e-5
 0.0010999999999999998
 0.011099999999999997
 0.05
u: 5-element Vector{Float64}:
 0.0
 9.999999999999997e-6
 0.00010999999999999995
 0.0011099999999999994
 0.004999999999999999

CC @ranocha

[efaulhaber]

A different, but probably more complicated solution could be to add a different kind of dtmin parameter, which is only used to terminate when automatic time integration chooses too small steps, but which doesn't terminate due to manually added tstops.

[oscardssmith]

I believe this was fixed (via different means) in https://github.com/SciML/SciMLBase.jl/pull/693. Specifically, on new SciMLBase, this gives

julia> solve(ode, Tsit5(), save_everystep=false, maxiters=1000, callback=cb).stats
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx┌ Warning: dt(1.3877787807814457e-17) <= eps(t)(0.1) , and step error estimate = 12.775142822640118. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).
└ @ SciMLBase ~/.julia/packages/SciMLBase/JUp1I/src/integrator_interface.jl:632

[efaulhaber]

Amazing, thanks!