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Germund Dahlquist established fundamental stability restrictions for linear multistep methods for ordinary differential equations. His 1963 paper analyzes A-stability and the associated order barrier, while identifying the special role of the trapezoidal rule. The result explains why increasing formal accuracy alone cannot guarantee stable integration of rapidly decaying differential equations.^[ref]

A linear multistep method approximates an ordinary differential equation $y^{\prime }=f(t,y)$ using values retained from several time levels. On a uniform grid $t_n=t_0+nh$, its constant-coefficient form isThe linearity is in the way values and derivatives are combined; the vector field $f$ itself can be nonlinear. The method is explicit when ${\beta }_k=0$ and implicit when the new value appears inside $f$ through ${\beta }_k\ne 0$. An implicit update requires a suitable algebraic solve, in addition to a formal recurrence.^[ref]

The two-step Adams-Bashforth method, for example, uses $Y_{n+1}=Y_n+h(3f_n-f_{n-1})/2$. It reuses the preceding derivative rather than calculating several new Runge-Kutta stages within one step. A $k$-step recurrence needs $k$ starting values, often supplied by a Runge-Kutta method of compatible accuracy.

Consistency controls the defect on a smooth exact solution; zero-stability controls the amplification of starting and local errors. Both are required by the Dahlquist equivalence theorem for convergence under its regularity assumptions. Changing the step sizes generally requires changing the coefficients and revisiting stability; uniform-grid coefficients cannot simply be reused at arbitrary time spacings.

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.