Error Estimate for a Finite-Difference Crank–Nicolson–ADI Scheme for a Class of Nonlinear Parabolic Isotropic Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Report on “Error estimate for a finite difference Crank-Nicolson-ADI scheme for a class of nonlinear parabolic isotropic systems’

A report for Mathematics

This study addresses the numerical solution of a class of nonlinear parabolic isotropic systems, which play a significant role in modeling phase transition processes such as solidification, Traditional numerical methods for such systems often face challenges in computational efficiency, accuracy, and stability. The authors propose a finite difference-based Crank-Nicolson-ADI scheme that effectively resolves these issues through the following key strategies:

1. Temporal discretization: The Crank-Nicolson method is applied to ensure second-order temporal accuracy.
2. Nonlinear term handling: The nonlinear term f(ф, u) is treated explicitly using extrapolation to approximate its values.
3. Spatial discretization: Finite differences are employed for spatial derivatives, guaranteeing second-order spatial accuracy.
4. Computational efficiency: The alternating direction implicit (ADl) technique decomposes the 2D problem into two lD problems (along xx- and yy- directions).significantly reducing computational complexity.

The authors provide a rigorous convergence analysis and validate their theoretical results through numerical experiments, the method's demonstrating effectiveness and convergence properties. Compared to traditional approaches, the proposed scheme achieves higher accuracy with lower computational demand, making it suitable for solving complex phase-field models. This work holds notable significance in computational physics. While the paper is well-structured and self-contained, minor revisions are recommended prior to publication.

Minor Revisions

1. Title: Remove the period at the end of the title. Suggested: “Error estimate for a finite difference Crank-Nicolson-ADI scheme for a class of nonlinear parabolic isotropic systems”
2. Abstract: The abstract lacks a clear statement of the study's unique contribution. Emphasize the improvements of the proposed method over existing approaches(e.g., enhanced stability, reduced computational cost, or higher-order accuracy).
3. Introduction (Page l):In the first sentence of the introduction, revise the phrase “frequently employed” to “commonly used" for academic tone:
4. Syntax and Grammar (e.g., Page 2): Please carefully review the sentence structure and grammar. Improve sentence structure in the following passage. Original: “These references include numerous interesting numerical computations and measurements of the efficiency of the underlying numerical technique. Suggested Revision: “These references include numerous numerical experiments and efficiency analyses of the proposed numerical technique.
5. Typographical errors (Page 18, Line 352): Correct “successive” to “successive” in.
6. Numerical Experiments (Page 17, Tables 1-2): Enhance clarity: Add descriptive titles to tables (e.g., “Table l:Errors and convergence rates of the Crank-Nicolson-ADI scheme in the discrete L, -norm”). Include a discussion on error sources (e.g., truncation errors from extrapolation, discretization limitations, or stability constraints).

Author Response

1. Title: Remove the period at the end of the title. Suggested: “Error estimate for a finite difference Crank-Nicolson-ADI scheme for a class of nonlinear parabolic isotropic systems”
2. We have removed the period
3. All other modifications have been taken care.

Reviewer 2 Report

Comments and Suggestions for Authors

1. The abstract needs to be cleaned up grammatically and shortened for clarity.
2. The CN ADI schemes are standard and not novel. The novelty of the application should be clarified upfront.
3. There are numerous typographical and grammatical issues, such as: a."nolinear" → should be nonlinear b.Sentences like “These models couple the energy (heat) equation for temperature with a nonlinear parabolic partial differential equation (p.d.e.) that includes another unknown, the phase, which takes values…” are repetitive.
4. The numerical experiments are minimal and not thoroughly discussed.
5. There is no comparison with other schemes, even simple ones like Euler or explicit methods, which would help contextualize the performance and advantages.
6. No discussion of runtime or computational efficiency, which would be helpful since ADI schemes are used partly for efficiency.
7. Reduce the similarity percentage from 21%.
8. Nearly 30% self-citation. The authors should cite appropriate foundational and latest references.
9. The manuscript needs thorough language editing to fix grammar, spelling, and clarity issues.

Comments on the Quality of English Language

There are numerous typographical and grammatical issues in the manuscript.

Author Response

All suggestions have been taken consideration.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Although the manuscript is of interest to a wide range of audience but certain polishing is essential.

1. The abstract has been included. 

2. Regarding the novelty of the method, the authors have not added any material.

3. Captions for two Tables are in the text. But there is only one Table in the text.

4. Regarding comparison with other schemes - No material added.

5. Regarding computational efficiency - No discussion / material added.

6. Two new references are added.

Author Response

 
 ( ) I would not like to sign my review report
(x) I would like to sign my review report
Quality of English Language
( ) The English could be improved to more clearly express the research.
(x) The English is fine and does not require any improvement.
Yes    Can be improved    Must be improved    Not applicable
Does the introduction provide sufficient background and include all relevant references?
( )    (x)    ( )    ( )
Is the research design appropriate?
( )    (x)    ( )    ( )
Are the methods adequately described?
( )    (x)    ( )    ( )
Are the results clearly presented?
( )    ( )    (x)    ( )
Are the conclusions supported by the results?
( )    ( )    (x)    ( )
Comments and Suggestions for Authors
Although the manuscript is of interest to a wide range of audience but certain polishing is essential.

1. The abstract has been included. 

2. Regarding the novelty of the method, the authors have not added any material.
 As we said in round1 the aim of the Paper is to demonstrate the merits of the ADI-ADI scheme theor4tically by proving the error convergence rate and confirming it by presenting indicative numerical results.

3. Captions for two Tables are in the text. But there is only one Table in the text.
I have corrected it

4. Regarding comparison with other schemes - No material added.
The following material was added : 
In the numerical solution of phase field models, the ADI-ADI order $\Delta t^2+h^2$  method demonstrates notable advantages over both the Euler-Euler order $\Delta t+h$ and Euler-ADI schemes order $\Delta t+h^2$. As detailed in Sfyrakis et al. \cite{ref16}, the ADI-ADI approach achieves second-order accuracy in both time and space while maintaining unconditional stability. This allows for larger time steps without compromising accuracy, leading to enhanced computational efficiency. In contrast, the fully explicit Euler-Euler method requires significantly smaller time steps to ensure stability, resulting in increased computational costs. While the semi-implicit Euler-ADI method offers improved stability over Euler-Euler, it still falls short in terms of accuracy and efficiency, particularly when addressing nonlinearities and higher-dimensional problems. Therefore, the ADI-ADI method provides a more robust and efficient framework for simulating complex phase transition phenomena. In table \ref{t4} they are shown the computing times (in seconds) required by the two schemes in a parallel implementation to achieve the error levels close to those shown in
table \ref{t1}   on one, two and four cpu's.

5. Regarding computational efficiency - No discussion / material added.
discussion and a table was added  with computing times for 1 2 and 4 cpus's

6. Two new references are added.

Submission Date
27 February 2025
Date of this review
29 Apr 2025 19:05:38

Author Response File: Author Response.pdf