The Study Of Geometric Algorithms For Hamiltonian Systems

Numerical integration methods have been indispensable for the research of Hamiltionian systems. However, the traditional integrators like Runge-Kutta algorithms will always destroy the invariability of the symplectic structure and conservative constant, which are caused by their artificial dissipation during the numerical integration. Therefore algorithms preserving geometric property take more and more attraction.Manifold correction methods and symplectic algorithms make up the two subclass of geometric algorithms for Hamiltonian system. Manifold correction methods stabilize the integrals of the motion by conrrection of numerical solutions. The main idea is that, numerically integrate the equations of the motion and then correct the currently obtained numerical solutions to pull the numerical path back to the true hypersurface, finally use the corrected ones as the initial conditions for the next step. While symplectic algorithms can preserve the symplcetic structure of the Hamiltonian systems, and then maintian the numerical errors of the first integrals only fluctuating on a small scale, not a linear increase. Both the two types of numerical methods are called geometric algorithms since they are involved in the geometric property of Hamiltonian systems.This paper just makes some advanced investigation of this two types of numerical method. Fistly, we generalize the velocity single scaling method and the velocity dule scaling method to the multiple ones, and successfully apply the scaling correction method to a two-dimensional discrete map. Secondly, by research of mixed symplectic algorithms for nonseparable Hamilton, we make the conclusion that Forest-Ruth algorightms of seven operators are not always accordance with Yoshida’s nine operator ones. Furthermore, a numerically determining method is introduced to obtain the order of a certain algorithm. Finally, we also discuss the application of the two mentioned geometric algorithms, and show that manifold correction methods are suitable for short-time’s quantitative computation while the other gives good performance in long-time’s qualitative analysis.