There are two measures of discretization error commonly used in discussing the accuracy of numerical methods for solving IVPs. The first is true or global error. For any , global error is simply the difference between the true solution and our numerical approximation to it:
Even though this is the error in which we are usually interested, it is a relatively difficult and expensive to estimate. The other measure of error is local error. It is the error incurred in taking a single step using a numerical method. If we let be the solution to the IVP,
then the local error at is given by
Most codes for solving IVPs estimate the local error at each step and attempt to adjust h accordingly. Control of local error controls global error indirectly; this, of course, depends on the stability of the problem itself. Most problems are at least moderately stable, and the global error is comparable to the error tolerance. Also, the cost of estimating global error is twice or more the cost of the integration itself.
Local and global errors at are illustrated graphically in Figure 1.
Figure 1: Local error (l.e.) and global error (g.e.).