Functionally fitted explicit pseudo two-step Runge–Kutta methods

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

“…Although EPTRK methods were originally developed for parallel computers (cf. [4]), it was shown in [14] that (F)EPTRK methods work better than one of the most frequently used Runge-Kutta methods the DOPRI45 method even in sequential computing environments on non-stiff problems. In [3], a class of explicit pseudo two-step Runge-Kutta-Nyström (EPTRKN) methods for solving non-stiff problem (1) was developed.…”

“…Similar to FEPTRK methods in [14], these methods have the advantage of integrating an ODE exactly if the solution of the ODE is a linear combination of functions in bases on which the FEPTRKN methods are developed. Using a similar collocation framework as in [14], we have shown in this paper that an s-stage FEPTRKN method has stage order p = s for any set of collocation parameters (c i )…”

“…For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]). Numerical experiments have shown that trigonometrically-fitted methods are superior to classical Runge-Kutta methods for solving ODEs whose solutions are periodic or nearly periodic functions with known frequencies (see, e.g., [8], [11], [12], [13], [14], [16], [17]). …”

“…In [14], a general class of functionally-fitted explicit pseudo two-step Runge-Kutta (FEPTRK) methods was developed. These methods can be considered generalized explicit pseudo two-step Runge-Kutta (EPTRK) methods.…”

“…These methods are developed to integrate an ODE exactly if the solution of the ODE is a linear combination of some certain basis functions (see, e.g., [2], [7], [8], [12], [13], [14], [17]). For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]).…”