Symplectic central difference scheme for quasi-linear autonomous Birkhoffian systems

Abstract

In this paper, a symplectic central difference scheme (SCDS) for the quasi-linear autonomous Birkhoffian system is proposed. Firstly, the definition of the quasi-linear autonomous Birkhoffian system in this paper is given. Then, by performing central difference discretization for the quasi-linear autonomous Birkhoffian equation, the SCDS is derived and proved to be strictly symplectic structure-preserving. Based on the precise integration method, an iterative process is used to obtain the transfer matrix of SCDS, which improves the accuracy of SCDS without significantly increasing the computational effort. In addition, the flexibility of SCDS for the quasi-linear autonomous Birkhoffian equation is discussed: the first is the applicability to odd-dimensional generalized Birkhoffian systems which appear widely in engineering problems; the second is the feasibility in structural dynamic response problems, for which the analytical procedure for applying SCDS based on the framework of the perturbation method is given. Several numerical examples verify the excellent performance of the proposed method.