phase-lag for the integration of second order periodic initial-value problems. II: Explicit method

In a recent paper (2) we gave a Noumerov-type method with minimal phase-lag for the integration of second order initial-value problems: y" = f(t, y), y(t,) = yO, y'(t,,) = yi. However, the method given there is implicit. We show here the interesting result that if the Noumerov-type methods of (2) are made explicit with the help of the classical second order method, then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size (l/40320) H6 and also a (slightly) larger interval of periodicity of size 2.75 than the phase-lag of size (1/12096)H6 and interval of periodicity of size 2.71 for the implicit method of (2). More interestingly, it turns out that Noumerov made explicit of Chawla (3) also has less frequency distortion than the (implicit) Noumerov method. (We shall assume a familiarity with the notation and discussion given in (2).)