A note on continuous-stage Runge-Kutta methods

We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK) methods, can give us a new perspective on RK discretization and it may enlarge the application of RK approximation theory in modern mathematics and engineering fields. A highlighted advantage of investigation of csRK methods is that we do not need to study the tedious solution of multi-variable nonlinear algebraic equations associated with order conditions. In this note, we will review, discuss and further promote the recently-developed csRK theory. In particular, we will place emphasis on geometric integrators including symplectic methods, symmetric methods and energy-preserving methods which play a central role in the field of geometric numerical integration.