Efficient energy-preserving integrators for oscillatory Hamiltonian systems

In this paper, we focus our attention on deriving and analyzing an efficient energy-preserving formula for the system of nonlinear oscillatory or highly oscillatory second-order differential equations q ″ ( t ) + Mq ( t ) = f q ( t ) , where M is a symmetric positive semi-definite matrix with M ≫ 1 and f ( q ) = - ∇ q U ( q ) is the negative gradient of a real-valued function U ( q ) . This system is a Hamiltonian system with the Hamiltonian H ( p , q ) = 1 2 p T p + 1 2 q T Mq + U ( q ) . The energy-preserving formula exactly preserves the Hamiltonian. We analyze in detail the properties of the energy-preserving formula and propose new efficient energy-preserving integrators in the sense of numerical implementation. The convergence analysis of the fixed-point iteration is presented for the implicit integrators proposed in this paper. It is shown that the convergence of implicit Average Vector Field methods is dependent on M , whereas the convergence of the new energy-preserving integrators is independent of M . The Fermi–Pasta–Ulam problem and the sine–Gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well-known Average Vector Field methods in the scientific literature.