Runge-Kutta Methods

Abstract

One-step methods (see Def. 2.1.8) form a particularly simple class of f. s. m. for IVP 1. Among these, a certain class of methods has commonly been associated with the names of C. Runge and W. Kutta and is widely used. These “Runge-Kutta methods” (RK-methods) are 1-step m+1-stage methods in the sense of Def. 2.1.10. However, only the final stage η ^m+1_v−1 of the value η_v−1 at t_v−1 enters into the computation of η _v ; also, only this final stage is normally taken as an approximation to the true solution. Thus, by formally disregarding the intermediate stages RK-methods may also be considered as 1-step 1-stage methods. We will use both interpretations depending on what is more convenient.