Continuous-stage Runge-Kutta-NystrÖm methods
This work provides a new theoretical framework for constructing RKN-type methods, which is significant for researchers in numerical integration of differential equations, but the results are primarily methodological and incremental.
The paper develops continuous-stage Runge-Kutta-Nyström (csRKN) methods by introducing a weight function, creating a larger framework that enables the use of weighted orthogonal polynomials to construct more effective RKN-type methods, including symmetric and symplectic integrators. Numerical experiments verify the effectiveness of the new integrators.
We develop continuous-stage Runge-Kutta-NystrÖm (csRKN) methods in this paper. By leading weight function into the formalism of csRKN methods and modifying the original pattern of continuous-stage methods, we establish a new and larger framework for csRKN methods and it enables us to derive more effective RKN-type methods. Particularly, a variety of classical weighted orthogonal polynomials can be used in the construction of RKN-type methods. As an important application, new families of symmetric and symplectic integrators can be easily acquired in such framework. Numerical experiments have verified the effectiveness of the new integrators presented in this paper.