The energy and group preserving schemes for multi degree of freedoms Duffing equations

In engineering and mathematics fields, the oscillatory problems of nonlinear oscillators are common problems. There are many computational methods which have been developed for solving the nonlinear oscillatory problems. However, most of these methods can not retain the energy. In this thesis, we develop a novel energy preserving scheme (EPS) for the undamped and unforced Duffing equation by recasting it to a Lie-type ordinary differential equation. The EPS can automatically preserve the total energy to be a constant value in a long term computation. Then, we will extend this problem to the damped and forced Duffing equations. Here, we use the group preserving schemes (GPS) to solve the problems, which can solve the problems effectively and accurately. Finally, we extend the problems to the coupled Duffing equations and three degrees of freedom Duffing equations. Also, we still can use the EPS and the GPS to solve the problems accurately. In each problem, we also compare the present results with the solution obtained by the fourth order Runge-Kutta (RK4) method, which has fourth-order accuracy. By comparing the EPS and RK4, we can see the advantages, accuracy and capability of preserving energy of the EPS.