General properties of integration methods

When one has recourse to a numerical algorithm, the main question is related to the errors which may arise. There are two kinds of error. 

The global error results from the addition of discretization and roundoff errors. Therefore, at least theoretically, there is an optimum step size for which the global error is minimum. However, for most computers and for stable algorithms (see below), in general, round-off errors may be neglected so that, essentially, discretization errors will be considered. 

A numerical method is said to be convergent if the global discretization error tends to zero when the step size tends to zero. The global discretization error is the difference between the computed solution (neglecting round-off errors) and the theoretical solution. Convergence is a minimal property of a numerical method and there is no use of a divergent method. Most numerical methods are convergent if they are consistent and stable. 

A numerical method is said to be consistent if the local discretization error (on one step) tends to zero when the step size tends to zero. In other words, more intuitively, the numerical algorithm tends to the mathematical equations as h - 0. The local discretization error is the error that would be made in one step if the previous values were exact and if there were no round-off errors. 

A method is stable if small variations of do not bring about 

A method is stable if small variations of r i do not bring about large variations of x , Xn + i. Thus stability indicates how both round-off and discretization errors are propagated for a sufficiently small step size. 

---