Finite Integration Method with Chebyshev Expansion for Shallow Water Equations over Variable Topography
, Ratinan Boonklurb 1,*,†, Lalita Apisornpanich 1 and Phiraphat Sutthimat 2,*,†Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsReview on the article “Numerical Algorithms for Shallow Water Equations Involving Variable Topography Based on Finite Integration Method with Chebyshev Polynomial Expansion” by Ampol Duangpan, Ratinan Boonklurb, Lalita Apisornpanich and Phiraphat Sutthimat
The article numerically solves the boundary value problem for partial differential equations (one- and two-dimensional). Differential equations are integrated over the spatial variable and approximated by Chebyshev polynomials, and difference schemes are used over the time variable. A linear algebraic system of equations is constructed to find a numerical solution. Numerical experiments are presented.
The results obtained by the authors develop and generalize numerical algorithms for solving PDEs by transforming differential equations into integral equations. Similar methods for other problems have been developed in previous works of the authors.
Main comments and questions to the authors.
- The literature review, especially the final part (lines 54-61), looks rather declarative. There is no analytical review and comparison of methods and results. In particular, it is rather uninformative to refer the reader to a large number of works at the same time (line 57). What is the difference between them? Where is the comparative analysis?
- Request to the authors to explain the references to the literature: a) [15] – this article considers a completely different problem (Burgers’ Equations); b) [16] – similarly (space-fractional differential equation); c) [17] – similarly (Volterra integro-differential equation); d) [18] – similarly (Nonlinear Poisson Equations). These publications contain applications of your method (integration using Chebyshev polynomials) to various problems. You do not refer to specific formulas or theorems. What are these references for? I believe that unnecessary and inadequate self-references should be removed, because there are already too many of them for a serious scientific article in a journal of this level.
- The article constructs a linear algebraic system of equations to find a numerical solution, but the issues of stability of the approximate solution, convergence of the method, and the order of accuracy of the method are not theoretically proven, but only refer the reader to the effectiveness of applying the method on numerical examples. I consider this a shortcoming of the work. Perhaps, at least some comments are needed for the general theoretical foundations of the developed algorithm? Several of the examples considered are interesting and illustrative, but this may not be enough.
Some technical remarks.
- Page 4. Formula between lines 85-86. Add a bracket to the first term of the vector on the left.
- Page 11. Lines 227, 228. Is a new notation used – CLF? (previously it was CFL).
I suggest the authors carefully check the text for technical typos.
The results of the work look new and from a practical point of view are important and confirmed by numerous examples. However, remarks 1-3 require responces from the authors and implementation of these responces in the text of the article. The work needs to be finalized.
Author Response
Comment 1: The literature review, especially the final part (lines 54-61), looks rather declarative. There is no analytical review and comparison of methods and results. In particular, it is rather uninformative to refer the reader to a large number of works at the same time (line 57). What is the difference between them? Where is the comparative analysis?
Response 1: We thank the reviewer for this valuable feedback. We have revised the manuscript to incorporate a more analytical discussion, as suggested. The new text on pages 2-3 (lines 46-72) now offers a comparative overview of existing numerical methods (FDMs, FVMs and high-order schemes) by discussing their respective strengths and weaknesses. This analysis provides a clearer rationale for introducing our proposed FIM-CPE method.
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Comment 2: Request to the authors to explain the references to the literature: a) [15] - this article considers a completely different problem (Burgers’ Equations); b) [16] - similarly (space-fractional differential equation); c) [17] - similarly (Volterra integro-differential equation); d) [18] - similarly (Nonlinear Poisson Equations). These publications contain applications of your method (integration using Chebyshev polynomials) to various problems. You do not refer to specific formulas or theorems. What are these references for? I believe that unnecessary and inadequate self-references should be removed, because there are already too many of them for a serious scientific article in a journal of this level.
Response 2: We thank the reviewer for this important suggestion. Our intent was to cite the body of work that establishes the development and versatility of our numerical method, the FIM-CPE, justifying its application here. However, we agree with the reviewer that the list of references could be more focused. Therefore, we have streamlined the references to retain only the most foundational citations for the method, as reflected on page 3, lines 64 and 72 of the revised manuscript. In addition, we have removed reference numbers [14], [15], [17], [19], [21], [22], [23] and [24].
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Comment 3: The article constructs a linear algebraic system of equations to find a numerical solution, but the issues of stability of the approximate solution, convergence of the method and the order of accuracy of the method are not theoretically proven, but only refer the reader to the effectiveness of applying the method on numerical examples. I consider this a shortcoming of the work. Perhaps, at least some comments are needed for the general theoretical foundations of the developed algorithm? Several of the examples considered are interesting and illustrative, but this may not be enough.
Response 3: We thank the reviewer for this important feedback and agree that a discussion on the theoretical foundations of our method would significantly strengthen the manuscript. In response, we have added two new dedicated subsections:
- Section 3.2 (page 11, lines 231-251) for the one-dimensional case.
- Section 4.2 (page 25, lines 484-497) for the two-dimensional case.
These sections now discuss the theoretical underpinnings of our FIM-CPE scheme, including how stability is maintained via the CFL condition and how accuracy is derived from the properties of Chebyshev polynomials and the finite integration approach. We also frame a full mathematical proof as a direction for future work. We believe these additions directly address the reviewer's concern.
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Comment 4: Page 4. Formula between lines 85-86. Add a bracket to the first term of the vector on the left.
Response 4: Thank you for spotting this typo. The closing bracket was indeed missing. We have corrected the equation, which is now located under line 106 on page 4 of the revised manuscript.
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Comment 5: Page 11. Lines 227, 228. Is a new notation used - CLF? (previously it was CFL).
Response 5: We appreciate you pointing out this inconsistency. This was a typographical error. We have corrected the typo to "CFL" in the revised manuscript, now located on page 12, lines 272-273.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsPlease see the attached pdf.
Comments for author File: Comments.pdf
Author Response
Comment 1: First, the title "Numerical Algorithms for Shallow Water Equations Involving Variable Topography Based on Finite Integration Method with Chebyshev Polynomial Expansion" is too long. The authors are advised to shorten it as much as possible.
Response 1: We agree that the title was lengthy and have shortened it in the revised manuscript to: "Finite Integration Method with Chebyshev Expansion for Shallow Water Equations over Variable Topography", as can be seen on the first page.
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Comment 2: There are too many unnecessary references in the manuscript. For example, on page 3, line 57, retain only reference [13] and delete the others. Similarly, on page 3, line 65, keep only reference [14] and remove the rest. Please review the entire manuscript accordingly.
Response 2: We have further streamlined the references throughout the revised manuscript to retain only the most essential citations that establish the method's foundations, as suggested. This is reflected on page 3, lines 64 and 79.
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Comment 3: What type of integral equations are solved in this work, and do they possess unique solutions? The authors must clarify this point in the manuscript.
Response 3: Thank you for this important question. The method transforms the governing PDEs into systems of linear Volterra integral equations of the second kind. We have added a clarification in our methodology section on page 8, lines 186-189, stating that the resulting discretized algebraic system has a unique solution, provided the system's matrix is non-singular.
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Comment 4: Why do the authors choose Chebyshev polynomials to solve the shallow water equations instead of other types of polynomials? Is there any specific advantage for using Chebyshev polynomials? The authors need to clarify and properly justify this choice in the revised manuscript.
Response 4: We have added a justification in lines 80-86 on page 3 of the revised manuscript. We chose Chebyshev polynomials for their minimax property, which minimizes approximation error, and because the use of Chebyshev nodes for collocation effectively mitigates Runge's phenomenon.
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Comment 5: Do the governing equations have a Green’s function as their fundamental solution? If so, using the fundamental solution could potentially improve the precision of the numerical results. The authors should address this point.
Response 5: This is an insightful point. We have added a brief note on page 3, lines 58-60 of the revised manuscript, explaining that while a Green's function exists for the linearized SWEs, a simple one is not available for the full nonlinear system, which motivates our direct numerical approach.
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Comment 6: Consider including a section demonstrating the convergence of the solution with respect to: (a) error analysis, and (b) panel resolution for numerical integration. This can be achieved by plotting figures that show error decay or the convergence of numerical estimates with increasing terms, which would add credibility to the results.
Response 6: We agree that a convergence study adds significant value. We have added a new subsection on page 17 (lines 336-351), 3.4 Convergence Analysis, which includes a figure and table demonstrating the error decay with an increasing number of nodes.
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Comment 7: A summary table comparing the key findings should be added to help readers quickly grasp the main takeaways.
Response 7: We have added a new summary table (Table 3) that consolidates the key findings from all numerical examples on page 28 (lines 528-530), to provide a concise overview of the algorithm's performance.
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Comment 8: In the conclusion, the authors should include a brief discussion on the limitations of the numerical algorithm, along with a short outlook on how it can be extended in future work. This would enhance the quality of the manuscript.
Response 8: Thank you for this advice. We have expanded the Conclusion section (page 28, lines 546-553) to more explicitly detail the current limitations of our method and to provide a clear outlook on future work.
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Comment 9: Since the authors have studied the approximate solution of integral equations by representing spatial terms using Chebyshev polynomials, they should also review other established methods for solving integral equations, particularly the Galerkin approximation, in the introduction section.
Response 9: We have revised the Introduction as recommended on page 3, lines 65-71. The new text now situates our FIM-CPE method with respect to other techniques for solving integral equations, including a direct comparison to the Galerkin method.
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Comment 10: Since this work concerns bottom topography, the authors are encouraged to include the following references in the introduction section of the manuscript, which address wave interactions over various bottom topographies: "Oblique wave diffraction by a bottom-standing thick barrier and a pair of partially immersed barriers" DOI: https://doi.org/10.1115/1.4055912
Response 10: We thank the reviewer for providing this relevant reference. We have included it in our Introduction to strengthen the discussion on wave interactions with bottom topography, on page 1 (lines 22-23) of the revised manuscript.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsMy comments have been taken into account. I propose to accept the article.
Reviewer 2 Report
Comments and Suggestions for AuthorsAll of my comments are appropriately addressed. I therefore recommend the manuscript be accepted in its current form.
