global (accumulated) error

The total error at a given point, after every step's own local error has compounded along the way.

The global error at step $n$ is $e_n = y(t_n) - y_n$ , the exact solution minus the computed approximation, after $n$ steps of accumulated local truncation error. A method is order $p$ when its global error scales as $O(h^p)$ -- one power of $h$ lower than its own local truncation error, since reaching a fixed endpoint takes $O(1/h)$ steps.

Examples

Euler's method (local error $O(h^2)$ ) has global error $O(h)$ -- Euler's method is called 'first-order' by its GLOBAL error, not its local one.