Implicit ERKN Methods For Second-order Oscillatory Differential Equations

A lot of problems arising in physical fields such as elastics, electric circuits, quan-tum physics, celestial mechanics etc., can be described by systems of ordinary differ-ential equations (ODEs). In this thesis, we are concerned with Runge-Kutta-Nystrom-type integrators for second-order initial value problems of ordinary differential equa-tions.It has been more than a hundred years since the classical Runge-Kutta methods were designed originally. The Runge-Kutta methods are mainly applied to solving first-order initial value problems of ODEs. A system of second-order ODEs can be turned into a system of first-order ODEs by adding a component of velocity and then can be solved by a Runge-Kutta method. The Runge-Kutta-Nystrom methods were designed by Nystrom to solve second-order ordinary differential equations directly. In1960’s Butcher set up the theory of rooted trees and B-series so that it is not difficult to obtain higher order Runge-Kutta methods. Accordingly, the theory of Nystrom-tree is established by Hairer et al.This thesis is divided into two chapters.Chapter1briefly presents the basic numerical methods of ordinary differential equations, including Runge-Kutta methods, RKN methods, Exponential Fitted RKN methods, Adapted Runge-Kutta-Nystrom (ARKN) methods.In chapter2, based on the work of the papers [9] and [8], implicit ERKN (ex-tended Runge-Kutta-Nystrom) methods for oscillatory second-order ordinary differen-tial equations are investigated. The order conditions for ERKN methods are restated and from which some implicit ERKN methods are constructed and analyzed. The results of numerical experiments for one-dimensional perturbed oscillators and multi-dimensional oscillatory second-order systems show the robustness and competence of the new methods in computation efficiency.