Local truncation error

Remarks

For an implicit method or a linear multistep method $\sum_{j=0}^{k} \alpha_j y_{n+j} = h \sum_{j=0}^{k} \beta_j f(t_{n+j}, y_{n+j})$, one inserts the exact solution in the same way: $\tau_{n+k} = \sum_j \alpha_j y(t_{n+j}) - h \sum_j \beta_j y'(t_{n+j})$. Some books divide by $h$ and call $\tau_{n+1}/h$ the truncation error; then order $p$ means $O(h^p)$. Example: for the Euler method, $\Phi(t, y, h) = f(t, y)$, and Taylor's theorem gives $\tau_{n+1} = y(t_n + h) - y(t_n) - h y'(t_n) = \frac{h^2}{2} y''(\xi_n)$, so the method is consistent of order $1$. Non-example: $\Phi(t, y, h) = 2 f(t, y)$ is not consistent, since $\tau_{n+1} = -h y'(t_n) + O(h^2)$ is not even $O(h^2)$. If $\Phi$ is Lipschitz in $y$ with constant $L > 0$ and $\abs{\tau_n} \le C h^{p+1}$ for all $n$, and if the method starts from $y_0 = y(t_0)$, then the global error satisfies $\abs{y(t_n) - y_n} \le \frac{C h^p}{L}\big(e^{L(t_n - t_0)} - 1\big)$.