= Explicit midpoint linear multistep method
{title2=$y_{i+1}=y_{i-1}+2h\Phi(t_i,y_i)$}
= Two-step midpoint method
{synonym}
For an <ordinary differential equation> $y'=\Phi(t,y)$, this <linear multistep method> uses a centered derivative across the two-step interval $[t_{i-1},t_{i+1}]$, evaluated at its midpoint $t_i$. It requires two starting values, $y_0$ and $y_1$. For a sufficiently smooth exact solution, <Taylor's theorem> gives
$$
y(t+h)-y(t-h)=2hy'(t)+\frac{h^3}{3}y'''(t)+O(h^5).
$$
The unscaled local residual is therefore $O(h^3)$ and the method has order two. Its first characteristic polynomial is $z^2-1$, with the simple roots $1,-1$, so it satisfies the <root condition for a multistep method>; the second root is a <parasitic amplification root>. This two-step method differs from the one-step <midpoint method> and the <implicit midpoint rule>.
For the special equation $G'=g(t)$ and $G(a)=0$, an even number $n=2r$ of steps gives $y_n=2h\sum_{j=1}^r g(t_{2j-1})$, independently of the odd-index starting value. It is the <composite midpoint quadrature rule> on $r$ intervals of width $2h$. An odd final index depends on $y_1$, which must be specified separately.
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