Euler Method - Global Truncation Error

Global Truncation Error

The global truncation error is the error at a fixed time, after however many steps the methods needs to take to reach that time from the initial time. The global truncation error is the cumulative effect of the local truncation errors committed in each step. The number of steps is easily determined to be, which is proportional to, and the error committed in each step is proportional to (see the previous section). Thus, it is to be expected that the global truncation error will be proportional to .

This intuitive reasoning can be made precise. If the solution has a bounded second derivative and is Lipschitz continuous in its second argument, then the global truncation error (GTE) is bounded by

where is an upper bound on the second derivative of on the given interval and is the Lipschitz constant of .

The precise form of this bound of little practical importance, as in most cases the bound vastly overestimates the actual error committed by the Euler method. What is important is that it shows that the global truncation error is (approximately) proportional to . For this reason, the Euler method is said to be first order.