Generalized explicit pseudo two-step Runge-Kutta-Nyström methods for solving second-order initial value problems
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A class of explicit pseudo two-step Runge-Kutta-Nyström (GEPTRKN) methods for 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 be considered a generalized version of the class of classical explicit pseudo two-step Runge-Kutta-Nyström methods. 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.
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