Implicit midpoint consistency bound ($\Vert {\tau }_n\Vert \le (M_3/24+L{M}_2/8)h^2$)

For an exact solution $y^{\prime }=f(t,y)$, define the normalized local truncation error of the implicit midpoint rule byIf $\Vert y^{\prime \prime }\Vert \le M_2$, $\Vert y^{\prime \prime \prime }\Vert \le M_3$ and the state Jacobian matrix of $f$ has operator norm at most $L$ on the intervening state segments, then the displayed bound holds uniformly on a fixed finite interval. To prove it, put $t=t_n+h/2$. Symmetric Taylor expansions giveFor the first bound, integrate the second-order remainder of $y^{\prime }(t+s)$ over the symmetric step; for the second, average the second-order remainders of $y$ at the endpoints. The odd terms cancel. The Lipschitz condition for $f$ and $y^{\prime }(t)=f(t,y(t))$ finish the proof by the triangle inequality. The unnormalized step residual is $h{\tau }_n=O(h^3)$.