= Explicit midpoint method
{title2=$y_{n+1}=y_n+h f(x_n+h/2,y_n+h f(x_n,y_n)/2)$}
This <explicit numerical method> is a two-stage <Runge-Kutta method> and a <one-step method>. For a sufficiently smooth <ordinary differential equation>, the <chain rule> gives $y''=f_x+f_yf$, and <Taylor's theorem> shows that its <local truncation error> is $O(h^3)$. Uniform <Lipschitz continuity> of $f$ in its solution argument gives an increment-function <Lipschitz constant> at most $L(1+hL/2)$. The <Lipschitz one-step error bound> then proves second-order <numerical convergence> on a fixed time interval. This method uses an intermediate prediction from $y_n$; the <explicit midpoint linear multistep method> instead requires a separate previous value $y_{n-1}$.
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