The Research And Comparison Of Continuous Finite Element Method Of Nonlinear Hamiltonian System

Hamiltonian system is the mechanical system which is used to describe dissipationless physical process and phenomenon,and it arise widely in the fields of physics,mechanics,engineering,pure and applied mathematics,etc.It is generally accepted that all real physical processes with negligible dissipation could be expressed,in some way,by Hamiltonian formalism,so that the research work of corresponding numerical methods is of great importance interest.Hamiltonian systems have two most important characteristics:conserved properties and preserve symplectic structures of flow.These characteristics can be maintained in the numerical calculation is of great significance.However,most numerical methods can not maintain the two properties simultaneously:symplectic and energy conservative in general(Ge-Masden theorem). Much of the recent research focus has been on symplectic,for instance, symplectic difference methods,symplectic RK methods,these methods can conserve symplectic properties well.But energy conservative is more important in many fields.So we turn to the finite element methods.The focus of the essay is on the The focus of the essay is on algorithms of the Kepler problem,which is a important example of nolinear Hamiltonian system.The Kepler problem has two conservation:Hamiltonian and angular momentum.When we evaluate numerical methods,there are three yardsticks: angular momentum and Hamiltonian conservation and small deviation in long time compute.We choose some representative methods from traditional algorithms,for example,symplectic difference methods,symplectic RK methods,etc.Then compared them with FEM from above three aspects.Our research shows that the finite element always conserve energy.For nonlinear Hamiltonian systems are proved high accuracyn conservate their angular momentum, the error of which of M Order Continuous Finite Element Methods is O(h^2m),and it is Time-independent.What is more,FEM can preserve longterm stability properties and precision of the calculated phase space track.It's orbital error is only one third of that of symplectic RK methods.