Order of a numerical method ($R_h=O(h^{p+1})$)

For a time-stepping formula written with an unscaled step residual $R_h$, order at least $p$ means that substituting a smooth exact solution gives $R_h=O(h^{p+1})$. Under the relevant stability of a numerical method assumptions and starting errors $O(h^p)$, this produces global error $O(h^p)$. Residuals divided by the step size instead use the equivalent convention $O(h^p)$.

The local truncation error is the defect left when exact solution values are inserted into a numerical update. For a one-step method started from the exact solution,A local error $O(h^{p+1})$ generally leads to global error $O(h^p)$ under a uniform stability bound.

For a time update with one-step defect ${\delta }_{h,k}$ after exact solution values are inserted, the normalized defect divides by the time step $k$. If ${\delta }_{h,k}=O(k^{p+1})$, the normalized local truncation error is $O(k^p)$. A spatial mesh contributes its own powers: for a diffusion update, ${\mathcal{T}}_{h,k}=O(k+h^2)$ means ${\delta }_{h,k}=O(k^2+k{h}^2)$. Under $k=O(h^2)$ these are respectively second and fourth powers of $h$. Neither normalization changes the actual order of a numerical method; it changes the power used to express its residual.