Integration Of Periodic Initial-value Problems Research Articles

An exponentially-fitted high order method for long-term integration of periodic initial-value problems

In this paper an exponentially-fitted eighth algebraic order explicit symmetric method is developed. The new method integrates exactly any linear combination of the functions {1, x, x 2, x 3, x 4, x 5, x 6, x 7,exp(± wx)}. Numerical results on long term integration of well-known periodic problems indicate that the new method is much more efficient than the “classical” symmetric eighth algebraic order developed by Quinlan and Tremaine [Astronom. J. 100 (1990) 1694–1700] and the well-known Runge–Kutta–Nyström Dormand et al. eighth algebraic order method [IMA J. Numer. Anal. 7(1987) 423–430].

Explicit high order methods for the numerical integration of periodic initial-value problems

Two explicit two-step hybrid methods of order 7 and 8 for the numerical integration of second order periodic initial-value problems are developed in this paper. The first of them has a large interval of periodicity and the other a minimal phase-lag. Each of them has seven stages per iteration. Numerical and theoretical results obtained for several well-known problems show the efficiency of the new methods.

A P-Stable Eighth-Order Method for the Numerical Integration of Periodic Initial-Value Problems

An eighth-order P-stable two-step method for the numerical integration of second-order periodic initial-value problems is developed in this paper. This method has seven stages per iteration and an interval of periodicity equal to (0, ∞); i.e., it is P-stable. Numerical and theoretical results obtained for several well-known problems show the efficiency of the new method.