Numerical Stability and Performance of Semi-Explicit and Semi-Implicit Predictor–Corrector Methods

Round 1

Reviewer 1 Report

The present form is good.

Reviewer 2 Report

I agree to the revision. Now this paper can be accepted.

Reviewer 3 Report

The focus of the proposed article is the problem of balancing the problems of efficiency and accuracy between explicit and implicit linear multistep methods and it is absolutely commendable that this balancing occurs through a rational two-step process. Having added, to testify to the effectiveness of the method, the differential systems (10), (11) and (12) it was good and increased the value of the article and its usability towards important problems encountered in applications. On my behalf the paper can be published

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The paper needs  major revision.

• Please cite relations (3), (4), (5) and (6).
• Please correct “matricesfor” in line 121.
• Please explain more about $A_11, …, A_22$ (line 134).
• The authors should mention to this work more carefully and should update some of the listed referenced in their paper in order to add a powerful for the paper. To help the authors in this direction I suggest the following references:

- https://doi.org/10.1016/j.rinp.2021.104456

- https://doi.org/10.1142/S0218348X22401235

• A numerical solution of variable order diffusion and wave equations, International Journal of Nonlinear Analysis and Applications, 12(1) (2021) 27-36.

Reviewer 2 Report

In this paper, the authors discuss the stability analysis of semi-explicit and semi-implicit predictor–corrector methods. The topic is interesting, but the writting is bad. In fact, there is not any theorem in the paper and there is no remarks to show the contriobution by comparing with some previous similar works. Also, there is no example to illustrate the theorey results.

Reviewer 3 Report

This article deals with the so called stability analysis of methods of discretization of systems of ordinary differential equations. In this regard, the authors would do well to explain, possibly in the introduction, which is moreover well articulated, that the term stability should not be understood in Lyapunov's sense, but that it refers to its numerical meaning. The investigation of new computationally efficient and stable numerical methods is, of course of great interest, due to the increasing complexity of meaningful mathematical models. This article focuses on the interesting problem of balancing the problems of efficiency and accuracy between explicit and implicit linear multistep methods. Such a balance is obtained first predicting the integration step by Adams-Bashforth method and correct it with Adams-Moulton method.

The different theoretical implementations are explained for the benefit of the reader in equations (1) and (2) Semi-explicit and semi-implicit ABM methods, (3) Semi-explicit ABM method, (4) Semi-implicit ABM method, (5) Semi-explicit AB-BDF method, (6) Semi-implicit AB-BDF method. Here too I would ask the authors to point out to the reader that at the end of the article there is a list of abbreviations, which in my opinion is very useful and positively qualifies the work. The stability of the proposed methods is examined in paragraph 2.2 and the following section 3, through appropriate numerical elaborations provide the stability regions of the semi-explicit, semi-implicit ABM and AB-BDF predictor-corrector methods. The concluding discussion is exhaustive and highlights the advantages of the proposed method, motivating the semi-implicit computation with the fact that each equation can be solved separately, so implicit algebraic equations have to be solved for one variable so that the computation of the full Hessian matrix in the Newton method is not required. In conclusion, the article has a relevant scientific foundation and is well argued, made the two minor adjustments that I have indicated, in my opinion it can be published