Variable-step global ODE error bound ($\Vert e_N\Vert \le e^{LT}\left(\Vert e_0\Vert +C{\sum }_n{h}_n^{p+1}\right)$)

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.