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A class of explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (GEPTRKN) methodsfor solving second-order initial value problems $y'' = f(t,y,y')$, $y(t_0) =y_0$, $y'(t_0)=y'_0$ has been studied. This new class of methods can beconsidered a generalized version of the class of classical explicit pseudotwo-step Runge-Kutta-Nystr\"{o}m methods. %The new methods will be denoted byGEPTRKN methods. We proved that an $s$-stage GEPTRKN method has step order ofaccuracy $p=s$ and stage order of accuracy $r=s$ for any set of distinctcollocation parameters $(c_i)_{i=1}^s$. Super-convergence for order of accuracyof these methods can be obtained if the collocation parameters $(c_i)_{i=1}^s$satisfy some orthogonality conditions. We proved that an $s$-stage GEPTRKNmethod can attain order of accuracy $p=s+2$. Numerical experiments have shownthat the new methods work better than classical methods for solving non-stiffproblems even on sequential computing environments. By their structures, thenew methods will be much more efficient when implemented on parallel computers.

Generalized explicit pseudo two-step Runge-Kutta-Nystr\"{o}m methods for solving second-order initial value problems