Global discretization error ($e_n=y_n-y(t_n)$)

The global error is the difference between the computed and exact solutions after accumulated numerical steps, rather than the local truncation error of a single step started from exact data. An estimate for the stability of a numerical method controls how initial errors and local defects accumulate. For a stable order-$p$ time integrator with starting errors $O(h^p)$, local defects $O(h^{p+1})$ accumulated over $O(1/h)$ steps give global error $O(h^p)$ on a fixed time interval.

If a one-step map is Lipschitz with factor at most $1+L{h}_n$ and its exact-start defect is at most $C{h}_n^{p+1}$, iteration of the error recurrence gives the displayed global bound on a time interval of length $T$. Since $\sum h_n\le T$, the defect sum is at most $T(\max h_n{)}^p$. Local tolerance control alone is not a global accuracy certificate: propagation, the number of steps, stiffness, roundoff and nonlinear-solve errors also enter. The discrete Gronwall inequality organizes this distinction.

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.