Generalized explicit pseudo two-step Runge-Kutta-Nyström methods for solving second-order initial value problems

Abstract

A class of explicit pseudo two-step Runge-Kutta-Nyström (GEPTRKN) methods for solving second-order initial value problems $y^{\prime \prime }=f(t,y,y^{\prime })$, $y(t_0)=y_0$, $y^{\prime }(t_0)=y_0^{\prime }$ has been studied. This new class of methods can be considered a generalized version of the class of classical explicit pseudo two-step Runge-Kutta-Nyström methods. %The new methods will be denoted by GEPTRKN methods. We proved that an $s$-stage GEPTRKN method has step order of accuracy $p=s$ and stage order of accuracy $r=s$ for any set of distinct collocation parameters $(c_i{)}_{i=1}^s$. Super-convergence for order of accuracy 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 GEPTRKN method can attain order of accuracy $p=s+2$. Numerical experiments have shown that the new methods work better than classical methods for solving non-stiff problems even on sequential computing environments. By their structures, the new methods will be much more efficient when implemented on parallel computers.

23 pages, 6 figures. arXiv admin note: text overlap with arXiv:1410.4090