Second Dahlquist barrier

The Second Dahlquist barrier states that an irreducible A-stable linear multistep method with constant real coefficients and only first-derivative evaluations has classical order at most two. Irreducible means that its characteristic polynomials $\rho$ and $\sigma$ have no common factor. The theorem concerns the entire recurrence, rather than a formula obtained by canceling unstable starting modes.^[ref]

The trapezoidal rule and BDF2 method attain order two while remaining A-stable. Increasing the number of retained values cannot produce a third-order method with all of these properties. A multiderivative multistep method lies outside the hypotheses: the A-stable third-order two-step multiderivative method uses second derivatives and therefore provides no contradiction. Likewise, high-order Gauss collocation methods are Runge-Kutta methods, a different class.