New Algorithm SharpFineRKN6 (see: https://github.com/SciML/OrdinaryDiffEq.jl/issues/677)

[HenryLangner]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• Any new documentation only uses public API

I have ran the JuliaFormatter so the formatting should be fine. And all code guidelines should be fulfilled however a short review would be appreciated just to make sure that the code follows all guidlines.

• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.

Additional context

New RKN Algorithm SharpFineRKN6 based on Some Nyström pairs for the general second-order initial-value problem.

I have attached a test file (test_SharpFineRKN6.txt, *.txt since *.jl is not supported for upload) for the Algorithm which does all the tests found in lib/OrdinaryDiffEqRKN/test/nystrom_convergence_tests.jl. The output is the following:

Some thoughts: All, besides three, tests pass. The first test which does not pass is the vector harmonic one. Here the l2 order is too low. The same phenomena can be observed for higher order RKN-Solvers (where machine precision is hit to fast. See: DPRKN12 big float test). Therefor I have added a big float test which passes. The second and third failing test is the adaptive IIP vs OOP one. However this is a phenomena already observed for this type of solvers see FineRKN4 adaptive IIP vs OOP tests. Nothing to worry about here. All in all everything looks as expected.

[HenryLangner]

The OrdinaryDiffEqRKN [QA] check failed with an Aqua error about not being able to locate ChainRulesCore this does not seem related to my changes however I can not re-run the test. Can someone do that?