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)=fq(t), where M is a symmetric positive semi-definite matrix with M≫1 and f(q)=-∇qU(q) is the negative gradient of a real-valued function U(q). This system is a Hamiltonian system with the Hamiltonian H(p,q)=12pTp+12qTMq+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.