 Order of a numerical method

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 $O(h^p)$.