More Runge-Kutta-Nyström Methods

[ranocha]

Here are some additional papers describing/developing/analysing Runge-Kutta-Nyström methods:

• [ ] Dormand, J. R., and PJi Prince. "New Runge-Kutta algorithms for numerical simulation in dynamical astronomy." Celestial Mechanics 18.3 (1978): 223-232.
• [ ] Dormand, J. R., M. E. A. El-Mikkawy, and P. J. Prince. "Families of runge-kutta-nystrom formulae." IMA Journal of Numerical Analysis 7.2 (1987): 235-250.
• [ ] Filippi, S., and J. Gräf. "New Runge–Kutta–Nyström formula-pairs of order 8 (7), 9 (8), 10 (9) and 11 (10) for differential equations of the form y ″= f (x, y)." Journal of computational and applied mathematics 14.3 (1986): 361-370.
• [ ] Fehlberg, E., S. Filippi, and J. Gräf. "Ein Runge‐Kutta‐Nyström‐Formelpaar der Ordnung 10 (11) für Differentialgleichungen der Form y′= f (x, y)." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 66.7 (1986): 265-270.
• [ ] Marthinsen, Arne. "Continuous extensions to Nyström methods for second order initial value problems." BIT Numerical Mathematics 36.2 (1996): 309-332.
• [ ] Fine, Jerry Michael. "Low order practical Runge-Kutta-Nyström methods." Computing 38.4 (1987): 281-297.
  • Fourth-order method
  • [x] Implementation https://github.com/SciML/OrdinaryDiffEq.jl/pull/1977
  • [ ] Dense output
  • Fifth-order method
  • [x] Constant step size: https://github.com/SciML/OrdinaryDiffEq.jl/pull/1948
  • [x] Error-based step size control #1976
  • [ ] Dense output
  • Dense output: Fine, Jerry Michael. "Interpolants for Runge-Kutta-Nyström methods." Computing 39.1 (1987): 27-42.
• [ ] Sharp, P. W., and J. M. Fine. "Some Nyström pairs for the general second-order initial-value problem." Journal of Computational and applied mathematics 42.3 (1992): 279-291.
  • This is derived to be more efficient than the methods of Fine mentioned above
• [ ] Sharp and Vaillancourt, New Nyström pairs for the general second-order problem
• [ ] Bettis, D. G. "A Runge-Kutta Nyström algorithm." Celestial mechanics 8.2 (1973): 229-233.
• [ ] Baker, T. S., J. R. Dormand, and P. J. Prince. "Continuous approximation with embedded Runge-Kutta-Nyström methods." Applied numerical mathematics 29.2 (1999): 171-188.
• [ ] Murua, A. "Runge-Kutta-Nyström methods for general second order ODEs with application to multi-body systems." Applied numerical mathematics 28.2-4 (1998): 387-399.
  • The abstract negates the title and states that there should be no dependence on the first derivative
• [x] #1495
• [ ] #865
• [ ] #112
• [ ] Check whether some common RK methods could take significant advantage from a specialized implementation instead of the general fallback mechanism?

[miguelraz]

Heads up to anyone fishing for a PR: I'm about to finish the first paper, would not advise redoing the tableaus fruitlessly as I did! :1st_place_medal: