High order explicit energy preserving Lie group method for some Hamiltonian systems

Abstract By the nonlinear transformation, some classical Hamiltonian systems can be written as the reformulation systems, which have the quadratic positive definite energy conserving function. These new reformulated systems are transformed into the ordinary differential equations on manifolds by the linear transformation. The explicit Runge-Kutta Munthe-Kaas (RKMK) method, which is a kind of Lie group method, is applied to solve the differential equations on manifolds. Therefore, a explicit energy preserving(EEP) RKMK method is proposed. The EEP-RKMK method is applied to solve the two classical dynamical systems: the pendulum problem and the Hénon-Heiles system. Thus, the explicit energy conserving schemes of the two classical dynamical systems are obtained. Numerical simulations investigate the effective of these new schemes in preserving energy conserving property of these equations and well simulating the dynamical behaviors. AMS subject classification: 37K05 65M20 65M70