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A Butcher order condition equates the coefficient assigned by a Runge-Kutta method to a rooted tree with the corresponding exact Taylor coefficient. Satisfying every condition through tree order $p$ is equivalent to Runge–Kutta order at least $p$.

The classical four-stage explicit Runge-Kutta method usesIt satisfies the fourth-order conditions for a Runge-Kutta method. Its polynomial stability function excludes A-stability, while $|R(iq){|}^2=1-q^6/72+q^8/576$ gives the imaginary-axis stability interval $|q|\le 2\sqrt{2}$.

A real vector field $f$ is dissipative in an inner product iffor all $x,y$. Two solutions of the ordinary differential equation $y^{\prime }=f(y)$ satisfyThus their distance never increases. The linear case $f(y)=Ay$ recovers the dissipative operator condition. B-stability asks whether a Runge-Kutta method preserves this contraction for every positive step size.

With $c=Ae$ and $C=\mathrm{diag}(c)$, a Runge-Kutta method has order at least four exactly when the eight Butcher order conditionshold; 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.