The Construction Of High Order Symplectic Integrator And Its Preservation Of Equilibria Structure

With an introduction on symplectic,algebraically stable,symmetric and someother interesting collocation methods given,we will answer two questions in this pa-per:1. How to construct high order methods with such characters?2.It is known that the symplectic methods should preserve the equilibria struc-ture. And we find the fact that, if the accuracy order of a SPRK method is increasedby symmetric composition,the structure-preservation region will be decreased accord-ingly.So how to get a composition method with larger suitable region?In the second chapter,the knowledge on W-transformation will be given.The col-location method with two parameters (δ,γ), R_s(δ,τ),will be discussed.And threenew types of methods, RadauI(γ), RadauII(γ), RadauIII(δ) ,which contain andextend the known Radau type methods,will be studied.Some examples are sup-plied.At the last of this chapter,we will continue to study Cs(δ,τ,α,β,γ) methods(which include Gauss,Radau,Lobatto type methods),and four new types methods,C_A,C_B,C_C,C_D ,are constructed and discussed.In the last chapter,we will give the idea of the preservation of equilibria structurefor symplectic integrators.New results are given. And we will give an introductionon algorithms with purely positive coefficients.It will be proved that the forward algo-rithms own larger suitable region than the known algorithms for structure-preservationthrough comparison.