local truncation error

The error a numerical ODE method introduces in a single step, if that step had started from an exact value.

For a one-step method advancing $(t_n, y_n)$ to $(t_{n+1}, y_{n+1})$ , the local truncation error $\tau_{n+1}$ is the error made by that ONE step alone, assuming $y_n$ was exact -- it isolates the method's own per-step approximation from any error already carried in from earlier steps. A method built by matching the true solution's Taylor expansion through the $h^k$ term has $\tau_{n+1}=O(h^{k+1})$ .

Examples

Euler's method has local truncation error $O(h^2)$ per step; RK4's is $O(h^5)$ -- three full orders better, per step, before any errors even accumulate.