ABSTRACT
Because the stability results of Chapter 5 were proven only for all “sufficiently small” step sizes, it is not clear whether an integration with a particular choice of step sizes will be stable. In this chapter we investigate stability in the context of a single integration. Specifically, if the solution sequence {y j} is the result of exact computation and {z j} is the result after some y m is perturbed slightly to z m, we ask whether the difference between y j and z j, grows as j increases. Stability in this sense is obviously very important in practice, but it is difficult to analyze. The same is true of the differential equation. To study the behavior of solutions with respect to a change in the initial values, we had to look at small changes. This restriction is not so bad, but we found that the behavior is specified in terms of the solution of a differential equation that itself depends on the solution of the original problem and as a consequence it is difficult to get any insight about stability. By restricting our attention to linear problems we could escape the limitation of small changes. By further restricting ourselves to quite special linear problems, the effects of changes could be worked out and useful guidelines about stability obtained. In this chapter we travel the same path for difference equations. There is an additional complication with difference equations due to the mesh. In the case of one-step methods, it is possible to work with a general mesh. In the case of methods with memory, we restrict ourselves to constant step size so as to simplify the task to the point that guidelines can be obtained. These guidelines are useful in practice because codes that select the step size automatically tend to work with a constant step size and it is often the case that there are sequences of steps of constant size to which the model applies. The results derived in this chapter are useful for selecting methods and when applied with care, can be useful in describing practical computation, but they are not all that one might hope for.
