the local truncation error

Imagine a perfect archer who, given the exact true position at the start of a step, fires one arrow with the method's formula. How far does that single arrow miss the true solution one step later, assuming everything before it was flawless? That one-step miss, measured from a perfectly correct starting point, is the local truncation error. It isolates the damage done by a SINGLE application of the method, with no accumulated history to muddy it.

Precisely, plug the EXACT solution y(t) into the method's update and see how badly the update fails to reproduce it over one step. For forward Euler, the update predicts y(t) + h*f(t, y(t)); the true next value is y(t+h). Taylor expansion gives y(t+h) = y(t) + h*y'(t) + (h^2/2)*y''(t) + ... and since y'(t) = f, the difference is about (h^2/2)*y''(t). So the local error per step is O(h^2): cut h in half and a single step's error drops by a factor of four. The leading power of h in this per-step error is what defines the method's order: forward Euler's O(h^2) local error makes it a first-order method (the order is one less than the local-error power, for reasons explained under global error).

The key honest point is the difference between LOCAL and GLOBAL error. Local truncation error is the mistake of ONE step in isolation; it looks small (O(h^2) for Euler). But you take roughly 1/h steps to cross a fixed time interval, and these errors accumulate and partly amplify, so the GLOBAL error you actually see at the end is one power of h worse, O(h) for Euler. Always ask which one a statement means: a method can have a flattering per-step error and a humbler end-to-end accuracy.