High-Accuracy P-Stable Methods with Minimal Phase-Lag for $y "= f(t, y) ^*$

Abstract

In this paper, we develop a one-parameter family of P-stable sixth-order and eighth-order two-step methods with minimal phase-lag errors for numerical integration of second order periodic initial value problems: $$ y''=f(t,y), \quad y(t_0)=y_0, \quad y'(t_0)=y'_0. $$ We determine the parameters so that the phase-lag (frequency distortion) of these methods are minimal. The resulting methods are P-stable methods with minimal phase-lag errors. The superiority of our present P-stable methods over the P-stable methods in [1-4] is given by comparative studying of the phase-lag errors and illustrated with numerical examples.