New 4(3) pairs diagonally implicit Runge-Kutta-Nystr?m method for periodic IVPs

摘要

New 4(3) pairs Diagonally Implicit Runge-Kutta-Nystr?m (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.