Stability And Order Of Variation Of Energy Of Several Symplectic Schemes

Almost all the real conservative physical processes,such as Celestial Mechanics, rigid body motion etc, can be cast in suitable Hamiltonian formulation. So the study of the Hamiltonian system is important, and it's numerical analysis would have a broad application.This article is divided into two parts. The first part discusses the variation of the energy. Conventional scheme of the differential equation doesn't consider the character of the physical process, such as energy preservation. Energy is an important concept of Hamiltonian system and in Hamiltonian system whose Hamiltonian function doesn't depend on time,the energy is a constant. But the energy may change with time after discretization. The author analyzed the order of variation of energy in symplectic and non-symplectic schemes when the Hamilton function has the second derivation. The analysis confirmed the expectation, for symplectic scheme behaves well.The second part is devoted to studying the stability of the scheme. The analysis of the stability of the scheme is important in numerical analysis. It decides whether the computation result is meaningful. Since the phase flow of the Hamiltonian system preserves the area of the phase space, the analysis of the stability of Hamilton system is different from the conventional schemes. The author analyzed the stability of the pade(2,2) symplectic scheme under the strong stability frame, and gives a necessity.At the end, the author computes an example of oscillations. Compared with the phase and energy variation graphs, it's clearly that the symplectic schemes behave well in preserving the energy and long time computation.