global truncation error

Local error is the slip of a single step under ideal conditions. But a real computation makes thousands of steps, each starting not from the true curve but from the imperfect point the last step left you on. Errors are inherited and added to. The global truncation error is the bottom line: the total gap between your computed value and the true solution at the final time, after all the steps have run and all their little mistakes have piled up and interacted.

Precisely, the global truncation error at a fixed time T is the difference y(T) - y_N between the true solution and the numerical value after N = (T - t0)/h steps. Here is the key relationship: to cover a fixed interval you need about 1/h steps, and each contributes a local error of roughly h^(p+1), so the total is about (1/h) * h^(p+1) = h^p. This is why the global error is one power of h smaller than the local error, and why we call a method 'order p' by its global behaviour: halving h cuts the global error by a factor of 2^p.

This is the number you actually care about, because it is the error in your final answer. It explains why high order pays off so handsomely: Euler (order 1) needs ten times the steps to gain one digit, while RK4 (order 4) gains roughly four digits for that same tenfold refinement. But the story is not monotone forever. As h shrinks toward zero, the step count explodes and round-off error — the unavoidable grit of finite-precision arithmetic — starts to grow and eventually dominates. There is a sweet-spot step size below which making h smaller makes the total error worse, not better.