Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge–Kutta methods of high order

This paper deals with the characterizations and construction of Poisson/symplectic and (φ⁻¹)-symmetric implicit high-order multi-revolution Runge–Kutta methods (MRRKMs). The basic tool is a modified W-transformation based on quadrature formulas and orthogonal polynomials. Two sufficient conditions can be obtained under which MRRKMs are Poisson/symplectic or (φ⁻¹)-symmetric. We construct two classes of high order implicit MRRKMs by using these sufficient conditions. Our results can be considered as an extension of related results of the standard Runge–Kutta methods in some references.