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 ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^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)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + 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.