A New Type of High-Order Mapped Unequal-Sized WENO Scheme for Nonlinear Degenerate Parabolic Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The work submitted for review is a development of the classical high-precision WENO scheme. The authors propose a new type of mapped unequal-sized weighted essentially non-oscillatory (MUSWENO) scheme for solving the nonlinear degenerate parabolic equation. Its content looks good and is ready for publication. However, it's worth considering making some minor improvements first, as discussed below.

1. I would recommend giving a broader range of applications of nonlinear degenerate parabolic equations in the Introduction.
2. In the Introduction the authors stop only at the application of nonlinear degenerate parabolic equations for gaseous medium. It is desirable to expand this description.
3. The references should be carefully edited and presented in accordance with the MDPI requirements. In particular, the first references (Line 38) are 7 and 21. The next references (Line 52) are 8 and 25. And so on.
4. I would also recommend to give details of variations and modifications of the WENO method with a more complete bibliographic reference.
5. It seems to me not quite successful numbering of mathematical equations, when each chapter of the paper begins with equation number 1. This can lead to confusion. It is more logical either to keep a continuous numbering or to put the number of the chapter before the number of the equation.
6. The description of Time Discretization Method should be improved. It is not quite clear how equation (35) is derived from equation (1). And then equation (36). I think that the reference to Ghosh and Baeder, 2012, is not sufficient.
7. The authors present a series of numerical experiments to evaluate the performance of the finite difference MUSWENO scheme. This is an interesting form of comparing MUSWENO with the classical sixth-order WENO scheme (WENO6). However, this comparison lacks consistency. There are 8 examples given, which do not show a single comparison criterion. While there is no great doubt about the results, I would recommend that a table similar to Table 1 (Example 3.1) be provided for each example. Or extend the data from this table to all examples.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript, a new scheme for solving the nonlinear degenerate parabolic equations is presented. I have the following concerns. Please address each of them in the manuscript.

 

Major comment

 

You have provided an explanation of previous research. But you have not provided the need for new research and the shortcomings of the research. Please explain to the reader.

 

 

 

Please revise Abstract. For example, this sentence should be in the introduction, not the abstract:” High-precision WENO schemes have been widely applied to solving various mathematical models in recent years.”

 

 

When a new method is proposed, some concepts should be presented about the characteristics of the method. I did not see anything like that in your manuscript. Please pay attention to it.

 

Edit the structure of the paper. It is not easy to read and follow it. Before Section 3, present an algorithm or framework of the proposed scheme.

 

 

Change the title of Section 4 to meet the journal's standards. This section as Conclusion of your manuscript is also weak and needs revision.

 

 

Minor comments

 

Labeling the axes in the figures have been forgotten. Please check them.

 

 

Please revise the captions of some figures because they are ambiguous, for example, in Figure 7, we have “. Contour and Surface. 200 × 200 grid points.”?

 

Please check the language of the manuscript, for example,

“distinct from the infinite propagation. speed of the classical heat equation”

“a new mpped unequal-sized WENO (MUSWENO) schemes”

“The results demonstrate that for smaller values of m, both schemes yield comparable numerical solutions.”

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript presents a new numerical scheme for solving a specific class of NDEs, namely nonlinear degenerate parabolic NDEs. The approach builds on the classical WENO framework by introducing a non-uniform-sized template strategy combined with newly designed nonlinear weights and mapping functions. This framework allows for an arbitrary choice of positive linear weights summating to unity, thus overcoming the common drawback of negative linear weights in high-order WENO reconstructions. The paper also rigorously analyzes the formal accuracy of the scheme and supports the proposed theoretical approach with a comprehensive set of numerical experiments.

The manuscript is clearly written, mathematically sound, and provides substantial numerical evidence to validate the proposed methodology. The work represents a valuable contribution to the numerical analysis of degenerate parabolic NDEs and would be useful to researchers in the fields of computational mathematics and applied physics.

Minor notes and suggestions for improvement:

1. Renumbering the formulas in the manuscript and reflecting this in the text. There are 3 formulas in the Introduction that are numbered. However, in the next section, the numbering starts again from the beginning.
2. A table of notations at the beginning of Section 2 may also be useful for readers who are not familiar with the WENO framework.
3. It may be useful to include one additional numerical example (examples) comparing the MUSWENO scheme with other recent approaches used by other researchers and cited in the manuscript. This is more of an idea for further study by the authors.
4. There are a few minor typographical errors that I assume will be corrected by the journal editors.

Overall, this is a strong manuscript that introduces a meaningful methodological improvement for the numerical solution of nonlinear degenerate parabolic equations. The theoretical formulation is well developed and the numerical tests convincingly demonstrate the advantages of the proposed scheme. In light of the above, I propose that the manuscript be published.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

I have no further comments.

Just revisit Comment #4 and use the past tense for the given sentences.

 

 

Author Response

Commons 1:[Just revisit Comment #4 and use the past tense for the given sentences.]

Response1:[Thank you for pointing this out. I agree with this comment. I have revised the paragraph you mentioned, changing the tense to the past tense.]