Runge–Kutta Methods

This chapter presents the graphs known as ‘rooted trees’ play a central role in the analysis of the accuracy of Runge-Kutta methods. To investigate the error in carrying out a single step of a Runge-Kutta method, one needs to compare successive terms in the Taylor expansions of the exact and the computed solutions. Having found the Taylor expansion of the exact solution to an initial value problem, one now find the corresponding expansion for the approximation computed by a Runge-Kutta method. Aim of comparing the Taylor expansions of the exact and computed solutions to an initial value problem will give an inconclusive answer unless the terms involving the various elementary differentials can be regarded as independent. The methods of Verner overcome the fault inherent in many of the Fehlberg methods, that the two embedded methods both have the same underlying quadrature formula.