Long term stability of numerical solutions of differential equations

The explosion in computer technology has led to thousands of new applications for the computer. Many of which require the efficient approximation of the solution to some governing equation. Unfortunately, the existing software for many applications is not adequate for the demands being placed upon it.

In this paper, we consider the problem of approximating the solution to an initial value problem. We focus on the behavior of an approximation scheme over a long time period and seek a technique that maintains a reasonable amount of accuracy over an arbitrarily long time period. Any algorithm which possesses this property will be referred to as "long time stable".