A new high precision energy-preserving integrator for system of oscillatory second-order differential equations

This Letter proposes a new high precision energy-preserving integrator for system of oscillatory second-order differential equations q″(t)+Mq(t)=f(q(t))q″(t)+Mq(t)=f(q(t)) with a symmetric and positive semi-definite matrix M   and f(q)=−∇U(q)f(q)=−∇U(q). The system is equivalent to a separable Hamiltonian system with Hamiltonian H(p,q)=12pTp+12qTMq+U(q). The properties of the new energy-preserving integrator are analyzed. The well-known Fermi–Pasta–Ulam problem is performed numerically to show that the new integrator preserves the energy integral with higher accuracy than Average Vector Field (AVF) method and an energy-preserving collocation method.

► A novel high order energy-preserving integrator AAVF-GL is proposed.
► The important properties of the new integrator AAVF-GL are shown.
► Numerical experiment is carried out compared with AVF method etc. appeared recently.