= Simpson multistep method
{c}
{title2=$Y_{n+2}-Y_n=\frac h3(f_{n+2}+4f_{n+1}+f_n)$}
Integrating a quadratic interpolant of the derivative over two steps gives this <linear multistep method>, which has order four. Its <local truncation error> is $-h^5y^{(5)}(t_{n+1})/90+O(h^7)$ 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.
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