Runge–Kutta–Fehlberg method

Remarks

Clause 1 is checked against Butcher's order conditions. There are $8$ conditions for the trees with at most $4$ vertices, for $b$, and $17$ for the trees with at most $5$ vertices, for $\hat b$. For instance $\sum b_i = \sum \hat b_i = 1$ and $\sum \hat b_i c_i^4 = \frac15$. Clause 3 follows from clause 1: the local errors of the two steps are $O(h^5)$ and $O(h^6)$, so their difference is the first one up to $O(h^6)$. The estimate drives adaptive step size control: a step is accepted when the estimate is below the tolerance, and the next step is scaled by about $(\mathrm{tol}/\mathrm{err})^{1/5}$. Erwin Fehlberg proposed such embedded pairs in 1969; in his original use the fourth-order value is propagated, while many codes propagate the fifth-order value.

Going deeper

For a student

This is the classic example of an embedded Runge–Kutta pair, often called RKF45. It was largely replaced in modern software by the Dormand–Prince method, whose coefficients are tuned to make the error of the fifth-order solution small and which reuses the last stage. The idea of embedding, two weight vectors on one set of stages, is the basis of nearly all explicit adaptive ODE solvers.