Normalized local truncation error (${\mathcal{T}}_{h,k}={\delta }_{h,k}/k$)

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.