Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq. (t)+ Bq(t) = f(q(t)), where A . .m.m is a symmetric positive definite matrix, B . .m.m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q) = ..qV (q) for a real-valued function V (q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q.,q)=12q.TAq.+12qTBq+V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.