A new approach to construct Runge-Kutta type methods and geometric numerical integrators

Abstract

We present a new approach to construct Runge-Kutta (RK) type methods and geometric numerical integrators including energy-preserving integrators, symplectic integrators and symmetric integrators. We first study RK methods with continuous stage introduced by Hairer based on orthogonal polynomial expansion and simplifying assumptions. Then s-stage RK methods will be obtained by using s-point quadrature formulae. We will present a unified framework to obtain energy-preserving integrators which covers some existing energy-preserving integrators. In addition, characterizations of symplectic and symmetric RK methods with continuous stage are presented. From these characterizations we obtain lots of symplectic and symmetric RK methods, which include and extend some known classical classes of symplectic and symmetric methods.