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

Abstract

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 0) = y 0, y′( t 0) = y′ 0. 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 ( 1 40320 )H 6 and also a (slightly) larger interval of periodicity of size 2.75 than the phase-lag of size ( 1 12096 )H 6 and interval of periodicity of size 2.71 for the implicit method [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].)