A major consideration in integrating differential equations is the numerical stability of the algorithm used. Since we have replaced the differential equation by a difference equation, we know that our results will not be exact. Discrepancies will arise when using different step sizes, for instance. This is the ``truncation error'' and depends on the method employed. Other errors that do not originate in the method correspond to the roundoffs performed by the computer, since it does not work with real numbers, but with a finite number of digits determined by the hardware. These roundoff errors will accumulate and can become significant in some cases.
In practice we determine the accuracy of our solutions by reducing the value of the step until the solutions unchanged at the desired level of accuracy.
In addition to accuracy, another important factor is the stability of the algorithm. For instance, it may occur that the numerical results are very good for short times, but diverge from the ``true'' solution for longer times. Such an algorithm is said to be ``unstable'' for the particular problem.