Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

Abstract

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 ∈ ℝm×m is a symmetric positive definite matrix, B ∈ ℝm×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 $$H(q', q) = \frac{1}{2}q'^{\rm{T}} Aq' + \frac{1}{2}q^{\rm{T}} Bq + V(q)$$ . 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.