Diagonal Implicit ERKN Method For Symplectic Oscillating Hamiltonian System

There is an important system which is Hamilton system in the power system.All real and dissipative negligible physical process can be expressed as Hamilton system whose range of application is very wide,including structural biology,pharmacology,semicon-ductor,superconductivity,plasma,celestial mechanics,materials and partial differential equations.Hamilton system is based on symplectic geometry.In order to keep the symplectic structure of Hamilton system,symplectic conditions and some novel explicit extended Runge–Kutta–Nystr?m(ERKN)methods with the order up to 4 are studied.Currently,the research to the problem of multiple frequency oscillation are staying on the explicit methods.However,implicit symplectic ERKN methods to solve the oscillation Hamilton systems have not been studied.Therefore,this paper studies diagonal implicit ERKN methods which improve the accuracy of numerical solution.This paper studies diagonal implicit symplectic ERKN methods for solving a oscilla-tory Hamiltonian system with the Hamiltonian function H(q,p)=1/2p~Tp+1/2q~TM_q+U(q).Based on symplectic conditions and order conditions,we construct some diagonal implicit symplectic ERKN methods.The stability of the obtained methods are discussed.Three numerical experiments are carried out and the numerical results demonstrate the remark-able numerical behavior of the new diagonal implicit symplectic methods when applied to the oscillatory Hamiltonian system.