Stabilizing Predictors for Weakly Unstable Correctors

should not be used for the numerical integration of y = /(x, y) if /„ < 0 along the true solution y(x) although the solution of ( 1 ) converges to y(x) for fixed finite x as h —» 0 (see, e.g., [1]). In fact, rapid oscillations, with an amplitude increasing exponentially as the numerical integration proceeds, will supersede the values approximating?/ (x) and eventually destroy the meaningfulness of the computation. This "weak unstability" occurring with (1) and similar algorithms has been well i analyzed (e.g., [1, p. 248 ff.]) and procedures have been suggested to weaken its effect (e.g., [2] ). We will show in this paper that it is quite easy to completely eliminate its cause : The combination of a judiciously chosen predictor with the weakly unstable corrector constitutes a strongly stable algorithm if the corrector is not iterated.