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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
- Implementation https://github.com/SciML/OrdinaryDiffEq.jl/pull/1977
- Dense output
- Fifth-order method
- Constant step size: https://github.com/SciML/OrdinaryDiffEq.jl/pull/1948
- 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.
- Fourth-order method
- 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
- #1495
- #865
- #112
- Check whether some common RK methods could take significant advantage from a specialized implementation instead of the general fallback mechanism?