Characterizations And Construction Of Symplectic And Symmetric Multi-revolution Implicit Runge-Kutta Methods Of High Order

This paper deals with the characterizations and construction of symplectic, (φ^-1)-symmetric, algebraically-stable and implicit multi-revolution Runge-Kutta (MRRK) methods of high order.Our methods are used for getting numerical solutions of problems which true solution are periodic or nearly periodic and have symplectic or (φ^-1) -symmetric structures. The main tool is a modified W-transformation based on quadrature formulas and orthogonal polynomials.Our main results are as follows:(1) The modified W-transformation is introduced. Using it we get a X-transfor-mation matrix by which the equivalent conditions of simplifying assumptions C_n(η) and D_n(ζ) are presented easily.(2) Sufficient conditions of symplectic,(φ^-1)-symmetric and algebraically-stable MRRK which are based on the modified W-transformation,are presented.(3) Using these sufficient conditions, we construct three classes of high order MRRK methods. The first class is symplectic,(φ^-1)-symmetric and algebraically-stable Gauss-Lobatto MRRK methods. The second class is symplectic and algebraically-stable Gauss-Radau MRRK methods.The third class is algebraically-stable MRRK methods. Finally, some numerical examples of MRRK methods are given.The above results (1) can be considered as extension of related results of Runge-Kutta methods obtained by Hairer and Wanner(1981)and so on. (2) and (3) can be regarded as extension of related results of Runge-Kutta methods obtained by Hairer and Wanner(1981),Sun(1993),Chan(1990),Grimm and Scherer(2003)and so on. Our methods of implicit symplectic ,(φ^-1)-symmetric,algebraically-stable and implicit multi-revolution Runge-Kutta(MRRK)methods of high order are,when N→ ∞, the classic methods of symplectic,symmetric,algebraically-stable and implicit Runge-Kutta methods of high order obtained by Hairer and Wanner(1981),Sun (1993),Chan(1990)and so on.