Is the predictor-corrector scheme A-stable?

Question

The following PECE scheme uses the Adam-Bashford method of order 2 as a predictor and the two-step BDF as the corrector. The two-step BDF scheme, on its own, is A-stable (stability region includes all of left half-plane). On the other hand, the Adam-Bashford method is not A-stable. What can we say about the A-stability of the combined scheme? My intuition is that the resulting PECE scheme is not A-stable because using the predictor makes it explicit and no explicit method is A-stable. If the PECE scheme is not A-stable, then what is the point of using it as opposed to just using, let's say, the A-stable BDF2 method by itself?

PECE scheme

Answer 1

The PECE method is an explicit method, as you already found out, and thus can not be A-stable from general principles.

If the implicit method is iterated out, then the A-stability of the whole scheme is the A-stability of the implicit method.

PECE is considered "stone-age". Modern implicit method implementations use a Newton-like or quasi-Newton method to solve the implicit equation. In this the same Jacobian matrix approximation is used over several steps.

The advantage of implicit methods is smaller coefficients in the error terms and a larger stability region even in the absence of A-stability (see $A(\alpha)$ stability and more detailed ideas). This in general allows to take larger steps.

The step size control of explicit method is dominated by the fastest changing component, even if this fast change happens below the absolute error tolerance. Violating this step size boundary leads to rapidly diverging numerical solutions.

With implicit methods such a violation of relative error bounds leads to a measurable increase in the relative error, but as long as that happens below the absolute error threshold this is insignificant. Look at discussions of explicit and implicit Euler to illustrate this general pattern.