Consider renaming `krylov_tol_abs` to `newton_tol_abs` and add `newton_tol_rel`

[ranocha]

The keyword argument krylov_tol_abs is passed as tol_abs to newton_krylov!, see

https://github.com/NumericalMathematics/Ariadne.jl/blob/9b8bfdf09f9cb7f6100dbe75cd818be897859124/libs/Theseus/src/dirk/dirk.jl#L254C72-L254C86

Ariadne.jl/libs/Theseus/src/imex/imex.jl

Line 258 in 9b8bfdf

algo = integrator.opts.algo, tol_abs = integrator.opts.krylov_tol_abs

I just realized while working with the code today that this is a bit confusing because it does not set an argument of the Krylov solver (:gmres by default), but of the overall Newton solver. See

Ariadne.jl/src/Ariadne.jl

Line 357 in 9b8bfdf

tol_abs = 1.0e-12, # Scipy uses 6e-6

Moreover, there is no option to pass tol_rel which may be handy.

[MarcoArtiano]

Actually that seems a bug or at least inconsistent with the rosenbrock implementation, where the variable krylov_atol and krylov_rtol are correctly used:

https://github.com/NumericalMathematics/Ariadne.jl/blob/main/libs/Theseus/src/rosenbrock/rosenbrock.jl#L50

https://github.com/NumericalMathematics/Ariadne.jl/blob/main/libs/Theseus/src/rosenbrock/rosenbrock.jl#L194-L199

Should we allow that flexibility also when the newton solver is used, i.e. having two sets of tolerances (one for the newton solver and one for the krylov solver)?

[ranocha]

The Krylov solver inside the Newton solver should determine the tolerances from the Eisenstat-Walker procedure (low accuracy when we are far away from the solution, stricter tolerances when Newton converges), so there should be no need to determine a fixed Krylov tolerance.