Functionally fitted Runge-Kutta-Nyström methods
This is an incremental theoretical extension of prior work on functionally fitted Runge-Kutta methods to the Nyström case, relevant for numerical analysts working on ODE solvers.
The authors extend a collocation framework to functionally fitted Runge-Kutta-Nyström methods, enabling exact integration of second-order differential equations whose solutions are combinations of certain basis functions, and achieve superconvergence under orthogonality conditions. Stability analysis is also provided.
We have shown previously that functionally fitted Runge-Kutta (FRK) methods can be studied using a convenient collocation framework. Here, we extend that framework to functionally fitted Runge-Kutta-Nyström (FRKN) methods, shedding further light on the fact that these methods can integrate a second-order differential equation exactly if its solution is a combination of certain basis functions, and that superconvergence can be obtained when the collocation points satisfy some orthogonality conditions. An analysis of their stability is also conducted.