 Residual bound for global ODE error

For a differentiable approximation $\eta$, put $d={\eta }^{\prime }-f(t,\eta )$. If $f$ is Lipschitz continuous with constant $L$ on the relevant region, the exact solution's error satisfies the displayed bound by Gronwall's inequality. Thus a bound on the residual over the whole interval supplies an a posteriori global error estimate. A small estimated defect at a single step does not furnish this bound by itself.