= Third-order conditions for an explicit Runge-Kutta method
For an explicit <Runge-Kutta method> with coefficients $A,b$, put $c=A\mathbf1$ and let $c^2$ mean componentwise squaring. The conditions $b^T\mathbf1=1$, $b^Tc=1/2$, $b^Tc^2=1/3$ and $b^TAc=1/6$ match the exact flow expansion through degree three: the two cubic differentials are $f^{\prime\prime}[f,f]$ and $(f^\prime)^2f$. For sufficiently smooth vector fields these give local error $O(h^4)$ and, under the usual finite-interval <Lipschitz continuity> assumptions, global error $O(h^3)$.
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