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A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. (English) Zbl 0565.65041

Authors’ summary: We consider a one-parameter family of Noumerov-type methods for the integration of second order periodic initial-value problems: \(y''=f(t,y)\), \(y(t_ 0)=y_ 0\), \(y'(t_ 0)=y'\!_ 0\). By applying these methods to the test equation: \(y''+\lambda^ 2y=0\), we determine the parameter of the family so that the phase-lag (frequency distortion) for the methods is minimal. The resulting method has a very small phase-lag of size \((1/12096)\lambda^ 6h^ 6\) (h is the step- size); interestingly, this method also possesses an interval of periodicity of size 2.71. Noumerov’s method has a phase-lag of size \((1/480)\lambda^ 4h^ 4\) and an interval of periodicity of size 2.449. The superiority of our method over Noumerov’s method is illustrated by two examples.
Reviewer: Z.Jackiewicz

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C25 Periodic solutions to ordinary differential equations
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References:

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