Gauss-Runge-Kutta-Nyström methods
The paper deals with an investigation of symplectic Runge-Kutta-Nyström (RKN) methods of the highest order. These methods are called Gauss-RKN methods as they are based on the Gauss-Legendre quadrature formulas. After a survey of RKN methods applied to Hamiltonian systems and a presentation of the conditions of symplecticness and symmetry, the authors prove the main results: Gauss-RKN methods are always symmetric, and symmetry is equivalent to symplecticness. Moreover, the authors show that for each stage number of all symplectic Gauss-RKN methods are given by a family of methods with one free parameter. In my opinion the paper is very important for further development of the theory of symplectic integrators and should be read by any specialist involved in study of numerical solutions of ordinary differential equations.
scientific article; zbMATH DE number 1158317