Functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström methods
This work provides a new class of numerical methods for solving second-order ODEs, offering improved accuracy for problems with solutions in specific function spaces, but is incremental as it generalizes existing EPTRKN methods.
The paper introduces functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström (FEPTRKN) methods for second-order initial value problems, proving that an s-stage method has step order p=s and stage order r=s, with superconvergence up to p=s+3 under orthogonality conditions. Numerical experiments show FEPTRKN methods outperform EPTRKN methods on problems where solutions are well approximated by the chosen basis functions.
A general class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge-Kutta-Nyström (EPTRKN) methods. We proved that an $s$-stage FEPTRKN method has step order $p = s$ and stage order $r = s$ for any set of distinct collocation parameters $(c_i)_{i=1}^s$. Supperconvergence for the accuracy orders of these methods can be obtained if the collocation parameters $(c_i)_{i=1}^s$ satisfy some orthogonality conditions. We proved that an $s$-stage FEPTRKN method can attain accuracy order $p = s + 3$. Numerical experiments have shown that the new FEPTRKN methods work better than do EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.