Dense output for FineRKN5

[HenryLangner]

Implementation of a specialized dense_output (6th order Hermite-Interpolation) for FineRKN5 derived in https://doi.org/10.1007/BF02307711 (for ref. view #677).

As shown in the convergence plots below, the Interpolation works fine (for In-Place ODEs, more on that below) and the dense_error is reduced significantly compared to the standard 3rd order Hermite-Interpolation:

In-Place (i) and Out-of-Place (o) convergence tests with and without dense error for the harmonic oscillator using the new dense output:

In-Place (i) and Out-of-Place (o) convergence tests with and without dense error for the harmonic oscillator using the standard dense output:

However testing the convergence with the new dense output for ODEs dependent on the first derivative (velocity dependent) results in breaking the Out-of-Place implementation in the sense, that the convergence worsen compared to the In-Place implementation. The difference can be seen in the plots below:

In-Place (i) and Out-of-Place (o) convergence tests with and without dense error for the damped oscillator using the new dense output:

I can not find the mistake in my Out-of-Place implementation.
Looking at the plots I am certain the mistake is not related to the coefficients since both implementations share the same ones and the In-Place on works regardless.

Any help is appreciated!

The code used for the convergence tests and plots is attached and can directly be pasted in the REPL for easy testing and replication of my results.
FineRKN5convergenceTests.txt

@ranocha

[ChrisRackauckas]

Can these be done lazily like is done with Vern?

[HenryLangner]

Yes it can be done lazily. As a matter of fact that was the idea I had in mind after getting the dense-output to work. I just did not know that the Vern solvers use that option already.