With $c=Ae$ and $C=\operatorname{diag}(c)$, a <Runge-Kutta method> has order at least four exactly when the eight <Butcher order conditions>
$$
b^Te=1,\quad b^Tc=\frac12,\quad b^Tc^2=\frac13,\quad b^TAc=\frac16,\quad b^Tc^3=\frac14,\quad b^TCAc=\frac18,\quad b^TAc^2=\frac1{12},\quad b^TA^2c=\frac1{24}
$$
hold; powers of $c$ are componentwise. These conditions match the numerical and exact coefficients indexed by <rooted trees> through order four. Matching the scalar <Dahlquist test equation> alone is insufficient to check the nonlinear <Butcher order conditions>.


 Fourth-order conditions for a Runge-Kutta method

ID: fourth-order-conditions-for-a-runge-kutta-method