Simpson multistep method ($Y_{n+2}-Y_n=\frac{h}{3}(f_{n+2}+4f_{n+1}+f_n)$)

Integrating a quadratic interpolant of the derivative over two steps gives this linear multistep method, which has order four. Its local truncation error is $-h^5{y}^{(5)}(t_{n+1})/90+O(h^7)$ and its zero-stability polynomial is $\rho (\zeta )={\zeta }^2-1$, with simple roots $1,-1$. The Dahlquist equivalence theorem therefore gives fourth-order convergence with sufficiently accurate starts. This attains the even two-step first Dahlquist barrier. Its undamped parasitic zero-step mode need not have favorable long-time absolute stability; finite-time convergence and stiff damping are separate requirements.