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For a smooth solution of $\Delta u=f$, the corrected formula ${\Gamma }_{9}u=h^{2}f+(h^2/12){\Gamma }_{5}f$ has unscaled residual $O(h^6)$ and normalized local truncation error $O(h^4)$. The inverse interior matrix has Euclidean operator norm $O(h^{-2})$; on $O(h^{-2})$ grid points this gives unweighted Euclidean nodal error $O(h^3)$. Uniform derivatives up to the boundary are required for a grid-independent remainder bound.

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.

A multiderivative multistep method uses higher time derivatives as well as values and first derivatives from several time levels. For an autonomous ordinary differential equation $y^{\prime }=f(y)$, the second time derivative is $y^{\prime \prime }=f^{\prime }(y)f(y)$ by the chain rule. Its local truncation error must include the higher-derivative terms. Consequently the usual second-order barrier for A-stable linear multistep methods, which only use first derivatives, does not apply.