Numerical methods for ordinary differential equations
Consistency and order

Suppose the numerical method is

The local (truncation) error of the method is the error committed by one step of the method. That is, it is the difference between the result given by the method, assuming that no error was made in earlier steps, and the exact solution:

The method is said to be consistent if

The method has order p {\displaystyle p} if

Hence a method is consistent if it has an order greater than 0. The (forward) Euler method (4) and the backward Euler method (6) introduced above both have order 1, so they are consistent. Most methods being used in practice attain higher order. Consistency is a necessary condition for convergence, but not sufficient; for a method to be convergent, it must be both consistent and zero-stable.

A related concept is the global (truncation) error, the error sustained in all the steps one needs to reach a fixed time t {\displaystyle t}. Explicitly, the global error at time t {\displaystyle t} is y N − y (t) {\displaystyle y_{N}-y(t)} where N = (t − t 0) / h {\displaystyle N=(t-t_{0})/h}. The global error of a p {\displaystyle p} th order one-step method is O (h p) {\displaystyle O(h^{p})}; in particular, such a method is convergent. This statement is not necessarily true for multi-step methods.
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