PRECISE SYMPLECTIC TIME FINITE ELEMENT METHOD AND THE STUDY OF PHASE ERROR

Hamiltonian system is one kind of important dynamical systems.Many kinds of symplectic methods were proposed for Hamiltonian systems, such as SRK, SPRK, multi-step method, generating function method and so on.The numerical methods for Hamiltonian can not be satisfied the character properties(symplectic and energy-preserving) at the same time.Recently, time finite element method was proposed for Hamiltonian systems, which was symplectic and could keep the conservation of energy.However, the mentioned methods have phase-drift(orbit deviation).For longtime simulation, the accuracy decay a lot.Precise Symplectic Time Finite Element Method is proposed for Hamiltonian systems(HPD-FEM), which could keep the conservation of Hamiltonian function and the structure of symplectic of Hamiltonian systems, as finite element method does.Meanwhile, HPD-FEM can highly decrease the phase error compared to FEM.HPD-FEM has fewer phase-error than the determined schemes aimed at decreasing the phase drift, such as:FSJS, RKN, and SPRK.FSJS, RKN and SPRK cannot keep the Hamiltonian function of Hamiltonian system, while HPD-FEM can keep the energy conservation.For the systems with different frequencies or the stiff systems, HPDFEM can simulate the signals both high and low frequency, with big time step.During the computation, HPD-FEM does not increase the cost of computation.Numerical experiment shows the validity of HPD-FEM.