A note on continuous-stage Runge-Kutta methods
For researchers in numerical ODEs and geometric integration, this provides a conceptual overview of csRK methods but is primarily a review without new results.
This note discusses continuous-stage Runge-Kutta methods for solving ODEs, highlighting their advantage of avoiding tedious order condition equations and emphasizing structure-preserving algorithms like symplectic and energy-preserving 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 stemming from order conditions. In this note, we will discuss and promote the recently-developed csRK theory. In particular, we will place emphasis on structure-preserving algorithms including symplectic methods, symmetric methods and energy-preserving methods which play a central role in the field of geometric numerical integration.