精细辛有限元方法及其相位误差研究

哈密顿系统是一类重要的动力系统,针对哈密顿系统,设计出多类辛方法:SRK、SPRK、辛多步法、生成函数法等.长久以来数值方法在求解哈密顿系统过程中辛特性和保能量特性不能得到同时满足,近年来提出的有限元方法,对于线性系统具有保辛和保能量的优良特性.但是,以上方法都存在相位漂移(轨道偏离)现象,长时间仿真,计算效果会大打折扣.提出精细辛有限元方法 (HPD-FEM)求解哈密顿系统,该方法继承时间有限元方法求解哈密顿系统所具有的保哈密顿系统的辛结构和哈密顿函数守恒性的优良特性,同时,通过精细化时间步长极大地减小了时间有限元方法的相位误差.HPD-FEM相较与针对相位误差专门设计的计算格式FSJS、RKN以及SRPK方法具有更好的纠正效果,几乎达到机器精度,误差为O(10-13),同时,HPD-FEM克服了FSJS、RKN和SPRK方法不能保证哈密顿函数守恒的缺点.对于高低混频系统和刚性系统,常规算法很难在较大步长下,同时实现对高低频精确仿真,HPD-FEM通过精细计算时间步长,在大步长情况下,实现高低混频的精确仿真.HPD-FEM方法在计算过程中精细方法没有额外增加计算量,计算效率高.数值结果显示本文提出的方法切实有效.

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.