In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows ar...In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called symplectic intoprators. Several symplectic methods are known in the class of Runge-Kutta methods. However, no higll order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are 'essentially' maintained by the discrete map, at least for linear problems.展开更多
文摘In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called symplectic intoprators. Several symplectic methods are known in the class of Runge-Kutta methods. However, no higll order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are 'essentially' maintained by the discrete map, at least for linear problems.
