Local Truncation Error

Local truncation error (LTE) is the error introduced in a single integration step, arising from truncating the Taylor series representation of the exact solution.

For the Explicit Euler method applied to $\dot{y}=g(t,y)$, the exact solution satisfies

$$y(t_{i+1})=y(t_i)+dt\cdot g(t_i,y_i)+O(d{t}^2)$$

so the local truncation error per step is $O(d{t}^2)$.

The LTE is one order higher than the Global truncation error because the global error accumulates the local error over all steps. If the integration interval has length $T$ and step size $dt$, then there are $N=T/dt$ steps, so

$$E_{\text{global}}\approx N\cdot E_{\text{local}}=\frac{T}{dt}\cdot O(d{t}^2)=O(dt)$$

This is why the Explicit Euler method is first-order overall, even though its local error is $O(d{t}^2)$.

Global truncation error | Order of a method | Explicit Euler method | Order of magnitude