Residual bound for global ODE error ($\Vert e(t)\Vert \le e^{L(t-t_0)}\Vert e(t_0)\Vert +{\int }_{t_0}^t{e}^{L(t-s)}\Vert d(s)\Vert ds$)

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.