Construction of Runge-Kutta type methods for solving ordinary differential equations

In this paper, we study Runge-Kutta methods with continuous stage, introduced by Butcher in 1987. By setting the coefficients of this family of methods and choosing the appropriate numerical integration, we derive new classes of Runge-Kutta methods. Furthermore, we extend the W-transformation technique by permitting W to be a non-square matrix. This allows us to construct more high-order implicit Runge-Kutta methods with some geometric properties. Specially, we provide Runge-Kutta methods with continuous stage which are (conjugate) symplectic, symmetric or energy-preserving for solving Hamiltonian systems. We also present some numerical experiments to verify our results.