A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems

摘要

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(t0) = y0, y′(t0) = y′0. By applying these methods to the test equation: y″ + λ2y = 0, we determine the parameter of the family so that the phase-lag (frequency distortion) for the method is minimal. The resulting method has a very small phase-lag of size 112096λ6h6 (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 1480λ4h4 and an interval of periodicity of size 2.449. The superiority of our method over Noumerov's method is illustrated by two examples.