Abstract
We will analyze in this paper some properties of unstable solutions in the numerical approximation of partial differential equations of the initial value kind, with spatial variable X and time t. It is often, although incorrectly believed that numerical instability always manifests itself by the appearance of components which are oscillatory in time as well as space with a period and a wavelength i.e. of the kind illustrated in Fig. 1. That this belief is widespread is probably due to the facts that: a) This type of instability is the most common, and b) Simple examples in the classical literature tend to emphasize this type only. But other types of numerical instability may and do occur. For instance, we demonstrate in §5 that instabilities of parabolic equations, which grow exponentially in time rather than in an oscillatory manner, exist with even-order accurate time-marching schemes. And in §6 we show that unstable solutions of hyperbolic equations have a wavelength of 4-AX, i.e. twice that of the parabolic case. We develop and illustrate these questions in detail by conducting an analysis in the frequency domain. In §8, we comment upon the difference between the notions of stepwise and point-wise stability, and illustrate certain types of unstable solutions, which occur despite the point wise stability of the scheme.