Consistency of a numerical method

For a one-step approximation $y_{n+1}=y_n+h\varphi (t_n,y_n,h)$ to an ordinary differential equation, insert the exact solution and define the unscaled local truncation errorConsistency of a numerical method requires ${\tau }_{n+1}/h\to 0$ uniformly on the finite interval being approximated, under the stated smoothness assumptions. Merely asking ${\tau }_{n+1}\to 0$ is too weak: the useless update $y_{n+1}=y_n$ has defect $h{y}^{\prime }(t_n)+O(h^2)$, which tends to zero but generally fails the scaled test. Here $O$ denotes Big O notation. MIT uses this division-by-step convention.^[ref]

For the Euler method, Taylor's theorem gives ${\tau }_{n+1}=h^2{y}^{\prime \prime }({\xi }_n)/2$. A bounded second derivative therefore gives local truncation error $O(h^2)$ and a scaled defect $O(h)$. Under a uniform Lipschitz continuity assumption on $f$ in $y$, within a region containing the exact and numerical solutions, numerical stability controls accumulation over about $T/h$ steps. This yields global error $O(h)$ with exact starting data. Consistency of a numerical method checks each step; numerical stability controls their combination.

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.