ContEvol Formalism: Numerical Methods Based on Hermite Spline Optimization

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see attachment below. 

Comments for author File: Comments.pdf

Author Response

General comments:

The article is devoted to ContEvol, a family of implicit numerical methods based on Hermite spline optimization.

The work is interesting and quite relevant, since the proposed method combines the advantages of implicit schemes with the technical simplicity of the problem of solving linear equations. The applicability of the method is demonstrated on classical problems, such as oscillations of a harmonic oscillator, some problems of celestial mechanics, and the construction of solutions to the Schrodinger equation.

The article is designed and structured in a standard way. It contains an introduction (motivation, connection with existing methods, main idea), a section on the harmonic oscillator, which demonstrates the methods of the first and second order and makes a comparison with the Runge-Kutta methods of the fourth and eighth orders, a section on classical problems of two and three bodies, in which the analysis of the conservation of energy and momentum is performed, as well as some numerical experiments are carried out. Next, the solution of the stationary Schrodinger equation for an infinite potential well, a harmonic oscillator and a Coulomb potential is performed.

General response:

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions highlighted in the re-submitted files.

 

Comments 1:

The main idea of the article is to use Hermitian splines to construct a numerical method in which the residual is minimized. However, it is worth explaining in more detail why Hermitian polynomials were chosen and not any other special functions. What clear advantage do they provide?

Response 1:

Thank you for this comment. I believe this confusion arises from the myriad of Charles Hermite's scientific contributions. While the technique is termed Hermite spline, the polynomials being used are common polynomials, not special functions like Hermite polynomials.

 

Comments 2:

It is worth noting that a huge amount of preparatory work must be done before the direct implementation of the numerical method. Formulas in the work are extremely cumbersome (e.g., (2.1.8), (2.1.10), (3.1.10)). Due to the increasing dimensionality of linear algebraic equation systems, the efficiency of calculations may decrease due to the need to calculate a large number of coefficients. Due to the abundance of long formulas and expressions, the article is difficult to understand. Some sections (e.g., quantum mechanics) appear to be overloaded with technical details at the expense of clarity of the main idea.

Response 2:

Thank you for pointing this out. This is a feature, not a bug, or the ContEvol formalism. The preparatory work is done so that the corresponding part of the problem-solving process does not need to be repeatedly done by computer. I believe the decreased efficiency due to increased complexity applies to all numerical methods and is not surprising. The mathematical derivation is necessary, and I have tried to maintain the clarity by providing intuitive interpretations. It would be greatly appreciated if concrete suggestions for further explanations could be provided.

 

Comments 3:

One of the significant drawbacks of the work is the lack of theorems on convergence, stability and estimation of the order of the method in the general case. The analysis is carried out mainly on model problems. There is a lack of research on sustainability. The question of the behavior of the method at large values of time is not resolved.

Response 3:

I appreciate this theoretical point of view. The incompleteness of the mathematical foundation was noted in Section 5. As stated in Section 1, "this work is principally for illustration and discussion of general strategies." I acknowledge that the research suggested by the reviewer would be a good follow-up to this practical proof-of-concept, but it is beyond the scope of the current work.

 

Comments 4:

In section 2.2, the author uses the RK8 coefficients from the Math Works website and discovers that the 4th order coefficients have an error. There is reason to believe that the error is not in the RK8 method, but in the calculations or in the author's use of incorrect coefficients. This is a serious inaccuracy that casts doubt on the thoroughness of the study.

Response 4:

Thank you for this comment. Recognizing that "there is reason," I had checked the results multiple times before submission. It would be appreciated if the specific error in this work could be pointed out.

 

Comments 5:

Table 1 compares the running time of methods in Python. However, the implementation of CE1, according to the author, uses ready-made expressions, while RK4 and Leapfrog are standard iterative algorithms. The comparison is incorrect. Since the fifth-order Taylor polynomials "hardcoded" into the code (in fact, CE1 in this case) should work faster than iterative RK4. and the advantage of using a high-order Taylor decomposition.

Response 5:

Thank you for this comment. All methods being compared are implemented as they usually are for practical problems, hence the comparison is appropriate. I do not understand the second half of this paragraph as the sentence is incomplete. The reviewer is welcome to suggest or conduct a comparison that they think is "correct."

 

Comments 6:

In Figure 3.2, it is stated that the CE1 method for the two-body problem is equivalent to the Taylor decomposition of the 5th order for the position and the 4th order for the velocity. But this means that all the cumbersome formalism with the minimization of functionality for this particular problem comes down to a precalculated Taylor decomposition. This calls into question the necessity of the entire ContEvol apparatus for problems where higher-order derivatives can be computed analytically.

Response 6:

Thank you for this comment. I recommend the reviewer read the penultimate paragraph of Section 3.2, which directly addresses this concern.

 

Comments 7:

The main drawback is that the algorithm is not strictly formulated, but a recipe is described, which requires for each new system of equations an analytical parametric form of the solution, obtaining the error functional, and analytically obtaining its derivative. The article mentions that a minimum of functionality can be used to adapt the step, but no specific scheme is proposed.

Response 7:

Thank you for this comment. This work presents a formalism, or a "recipe," not an algorithm. The step length adaptation is informed by the relationship between the error and the step length, and is exemplified after the idea of adaptation is mentioned.

 

Comments 8:

Recommendation:

1. Strictly and clearly indicate the boundaries of the applicability of the method.
2. Recognize its equivalence to the Taylor decomposition in the simplest cases.
3. Demonstrate its real advantage on a problem where the classical Taylor decomposition is impossible or inefficient (if such problems exist). Or to prove that there are no such tasks.

Response 8:

Thank you for these recommendations. For 1, a dedicated subsection on the limitations of the methodology has been added to Section 5. For 2 and 3, I do not know what the "classical Taylor decomposition" is. If it refers to Tayler expansion, please see Response 6. The "deep Taylor decomposition" method is relatively new and not directly related to this work.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript introduces a novel family of numerical methods, termed ContEvol, based on Hermite spline optimization, and validates its effectiveness on prototype cases including the harmonic oscillator, the two-body problem in celestial mechanics, and the stationary Schrödinger equation in quantum mechanics. The research presented is highly innovative, supported by rigorous theoretical derivations and comprehensive numerical experiments, demonstrating the potential of the ContEvol method in terms of accuracy, efficiency, and adherence to physical conservation laws. The manuscript is well-structured and generally well-written. I recommend acceptance after minor revisions.

To further enhance the quality of the manuscript, the following points should be addressed:

1. Introduction and Literature Review:
   The literature review in the introduction could be strengthened. It is suggested to briefly mention one or two classic implicit methods (e.g., Backward Euler) or spline collocation methods when discussing the categories of numerical methods. This would provide a clearer contrast and more effectively highlight the core innovation of ContEvol—solving only linear systems of equations.
2. Language and Formatting:

• Minor typographical errors were found and require careful proofreading throughout the manuscript. For instance, "ContEval" on Page 5 should be corrected to "ContEvol".
• Some sentences are overly complex. Breaking them down is recommended to improve readability and flow.

3. Originality and Depth of Analysis:
   The manuscript excels in demonstrating its originality and analytical depth. To enhance scholarly rigor, it is recommended to add a dedicated subsection on "Limitations and Future Work"in the Conclusion or Section 5. This subsection should systematically consolidate the limitations mentioned throughout the text (e.g., lack of strict symplecticity, moderate benefits of higher-order methods, challenges in handling infinite boundaries).
4. Figures and Numerical Experiments:
   While Section 3.5 provides a formal description for the three-body problem, it lacks a concrete numerical example. It is highly recommended to include a numerical experiment for a three-body problem, accompanied by an orbital diagram. This would visually demonstrate the method's effectiveness and scalabilitybeyond the two-body problem, significantly strengthening the manuscript's conclusions.
5. References:
   It is advised to supplement the introduction and relevant discussions with citations from the last five years concerning high-order geometric (structure-preserving) integratorsand machine learning-based solvers for differential equations. This will help to more precisely contextualize the present work within the current state-of-the-art and further underscore the innovative contributionof the ContEvol method.

Author Response

General comments:

This manuscript introduces a novel family of numerical methods, termed ContEvol, based on Hermite spline optimization, and validates its effectiveness on prototype cases including the harmonic oscillator, the two-body problem in celestial mechanics, and the stationary Schrödinger equation in quantum mechanics. The research presented is highly innovative, supported by rigorous theoretical derivations and comprehensive numerical experiments, demonstrating the potential of the ContEvol method in terms of accuracy, efficiency, and adherence to physical conservation laws. The manuscript is well-structured and generally well-written. I recommend acceptance after minor revisions.

General response:

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions highlighted in the re-submitted files.

 

Comments 1:

To further enhance the quality of the manuscript, the following points should be addressed:

1. Introduction and Literature Review:
   The literature review in the introduction could be strengthened. It is suggested to briefly mention one or two classic implicit methods (e.g., Backward Euler) or spline collocation methods when discussing the categories of numerical methods. This would provide a clearer contrast and more effectively highlight the core innovation of ContEvol—solving only linear systems of equations.

Response 1:

Thank you for this insightful suggestion. I have revised the introduction to mention these methods.

 

Comments 2:

1. Language and Formatting:

• Minor typographical errors were found and require careful proofreading throughout the manuscript. For instance, "ContEval" on Page 5 should be corrected to "ContEvol".
• Some sentences are overly complex. Breaking them down is recommended to improve readability and flow.

Response 2:

Thank you for pointing this out. I have carefully proofread the manuscript and revised it to reduce typos and improve clarity.

 

Comments 3:

1. Originality and Depth of Analysis:
   The manuscript excels in demonstrating its originality and analytical depth. To enhance scholarly rigor, it is recommended to add a dedicated subsection on "Limitations and Future Work"in the Conclusion or Section 5. This subsection should systematically consolidate the limitations mentioned throughout the text (e.g., lack of strict symplecticity, moderate benefits of higher-order methods, challenges in handling infinite boundaries).

Response 3:

Thank you for this recommendation. I have added such a dedicated subsection.

 

Comments 4:

1. Figures and Numerical Experiments:
   While Section 3.5 provides a formal description for the three-body problem, it lacks a concrete numerical example. It is highly recommended to include a numerical experiment for a three-body problem, accompanied by an orbital diagram. This would visually demonstrate the method's effectiveness and scalabilitybeyond the two-body problem, significantly strengthening the manuscript's conclusions.

Response 4:

Thank you for this suggestion. I agree that a concrete three-body example would be a great addition. However, I think two-body examples are sufficient to demonstrate the viability of methods proposed in this work for initial value problems and would prefer to leave more complex problems to future work.

 

Comments 5:

1. References:
   It is advised to supplement the introduction and relevant discussions with citations from the last five years concerning high-order geometric (structure-preserving) integratorsand machine learning-based solvers for differential equations. This will help to more precisely contextualize the present work within the current state-of-the-art and further underscore the innovative contributionof the ContEvol method.

Response 5:

Thank you for this insightful comment. I have mentioned the geometric integration theory in the subsection on limitations. While ML-based solvers are valuable, I think the training set needs to be supplied by non-ML methods, therefore ML may be out of scope.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you for the comments. And for correcting the text. There are no additional comments.