A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I enjoyed reading the article from Prof. Ganapol. I suggest publishing this work in this journal. I only have a minor comment for the author. Currently, the number of references are too limited. I suggest adding more recent references to the work, to show this is an important current work.

Author Response

1. Currently, the number of references are too limited. I suggest adding more recent references to the work, to show this is an important current work.  Revised: I have included several more references. See revision 1.

Reviewer 2 Report

Comments and Suggestions for Authors

In this work, 

the author explored numerical and analytical solutions for the transport of neutrons in 1+1 space-time. I think that the paper contains potentially interesting results with possible applications in model for neutron propagation. However, there are several unclear points to me: i) I notice a very limited set of references in the bibliography while of course this is a largely studied subject. I suggest to the author to extend the bibliography to improve the clarity of the exposition and the relations with previous works. ii) It is not clear if the solutions are completely novel, in some parts they just look like standard solutions using Legendre polynomials. What is the main novelty of this work compared to previous solutions? iii) It is not clear from the caption where these error ratios in Fig.7b are coming from, more explanations are needed. iV) From the physics prospective, What is the the typical temperature e kinetic energy where these methods can be applied? Can it be useful to study cold neutron propagation in long-baselines (see for example arXiv:2006.04907 for the European Spallation Project for neutron base-lines)? I will reconsider this paper for the publications after systematic replies to these issues. 

Comments on the Quality of English Language

The English quality is sufficiently good. 

Author Response

i) I notice a very limited set of references in the bibliography while of course this is a largely studied subject. I suggest to the author to extend the bibliography to improve the clarity of the exposition and the relations with previous works.  Response: I have included additional references for the mathematics and for different solutions and an Appendix of previous solutions.

 ii) It is not clear if the solutions are completely novel, in some parts they just look like standard solutions using Legendre polynomials. What is the main novelty of this work compared to previous solutions? Response: All solutions to the PN equations have the same form if orthogonality is the method of solution.  The novelty here is how the moments are obtained. There are no other solutions that solve the second order or parity form for the moments. I have now included:

Response: It should be stated that the approach taken thus far is not unique.  The first order ODE, Eq(10a), can be solved in terms of exponential eigenfunctions [7,8,9].  Here our approach is very different, where we solve a set of second order ODEs for , Eq(14c) in terms of normalized eigenfunctions.  Instead however, one could equally derive a set of second order ODEs for psi , but there are additional technical issues associated with this second approach, and therefore will not be further considered.

 iii) It is not clear from the caption where these error ratios in Fig.7b are coming from, more explanations are needed.  Response: The following is in the text

The effectiveness of W-e acceleration is shown by Fig. 1, which is the ratio of the relative error with and without acceleration over all directions at the seven spatial coordinates.  One observes the W-e relative errors at convergence are generally smaller than the errors of the original sequence for this benchmark solution. This is further confirmation of the high quality-- to one unit in the seventh and nearly one unit in the eighth place-- of the proposed DPN algorithm.  Over half of the fluxes converged by W-e showing the significance of the Wynn-epsilon algorithm.

 iV) From the physics prospective, What is the the typical temperature e kinetic energy where these methods can be applied? Can it be useful to study cold neutron propagation in long-baselines (see for example arXiv:2006.04907 for the European Spallation Project for neutron base-lines)?  Response:  This is for monoenergetic neutrons so they are unaffected by the temperature associated with the nuclei from which they scatter.  If fission is included, then this kind of analysis could be for thermal neutrons, with some adjustments after they have slowed down from fission energies ~1Mev.  As to the spallation project, I see no direct application as I am only doing classical physics with the transport equation. One can however do neutrino physics by adding an appropriate scattering term and spin-spin coupling. There is also the possibility to use neutron transport for detectors. Very interesting project.  Thanks for pointing it out.

 

 

 

 

 

Reviewer 3 Report

Comments and Suggestions for Authors

I only have minor comments. Very well written paper.

Page 4 line 129.

“Expressing the RHS more conveniently in terms of Psi(x) gives”.

I don’t see Psi(x) in the following mathematical relation. This term does emerge in section 2.2. Am I missing something?

 

Eqs (31a,b)

Seems that the hat symbols above A are shifted slightly.

I have noticed that this happens in other places and suggest checking for consistency.

Author Response

i. Page 4 line 129.

“Expressing the RHS more conveniently in terms of Psi(x) gives”.

I don’t see Psi(x) in the following mathematical relation. This term does emerge in section 2.2. Am I missing something?

Response:  No, I fixed the reference

ii. 

Eqs (31a,b)

Seems that the hat symbols above A are shifted slightly.

I have noticed that this happens in other places and suggest checking for consistency.

Response: That is what WORD gives. Can't do anything about it.

Reviewer 4 Report

Comments and Suggestions for Authors

The idea of the double PN benchmark solution is suggested in the paper titled “Double PN Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry”. The manuscript is not written and formatted well. Some of the suggested points below contradict its quality. 

1. The abstract does not include the final output values. Please, rewrite the abstract.

2. The introduction was brief, it needs to be rewritten to include a clearer motivation and problem statement. A satisfactory literature review should rewrite with some information about the works.

3. Could you summarize the proposed DPN approximation algorithm in steps?

4. Check and revise Eq. 4

5. To improve the overall quality of the paper, it is essential to well-written the manuscript including the English language as well as the punctuation mistakes.

6. Recent references must be added.

Comments on the Quality of English Language

Editing of English language required.

Author Response

1. The abstract does not include the final output values. Please, rewrite the abstract

Response: Now included in abstract:

In comparison to a well-established fully analytical response matrix discrete ordinates solution RM/DOM benchmark using an entirely different method of solution for a non-absorbing 1mfp thick slab with both an isotropic and beam source, the DPN algorithm achieves nearly 8 and 7 place precision respectively.

2. The introduction was brief, it needs to be rewritten to include a clearer motivation and problem statement. A satisfactory literature review should rewrite with some information about the works.

Response: Introduction is now:

1. Introduction is expanded to

Boltzmann's equation of particle transport, indeed presents a significant challenge and noteworthy opportunities to solve because of its complexity and wide range of physical phenomena it describes [1,2].  Originally, the non-linear integro-differential equation, as prescribed by kinetic theory of particle motion seemed unsolvable.  With time however, and advances in mathematics and physical applications, where, in some cases, non-linearity could be relaxed to give a linear equation, the situation changed [3].  In the early to mid twentieth century, a flurry of analytical solutions were constructed for the linear and linearized Boltzmann equation primarily based on solving partial differential equations (PDEs) with distributions admitted, specifically for one-dimension.  Alongside the development of analytical solutions were numerical solutions as well such as Monte Carlo and discrete ordinates methods, thus, coupled with increasing computational performance, enabled practical use of Boltzmann’s equation in nuclear reactor design.  With numerical solutions and applications to particle physics, there arose a need for numerical method’s verification, which required the development of benchmarks and benchmarking.  This, in turn, led to a host of numerical benchmark solutions to more relevant transport applications requiring sophisticated benchmarking techniques but still generally limited to model problems.  Some readers may have the misconception that benchmarking comprehensive transport algorithms is a wasted exercise since only idealized cases hold; however, the opposite is true.  Even the most advanced numerical method to solve the transport equation can contain undetected errors that a benchmark, even for a simple problem, can easily find.  The following presentation is concerned with developing a high precision (DPN) benchmark in one-dimension as another of the author’s high-precision benchmarks. The Appendix summarizes three previous benchmarks to provide a backdrop upon which to compare the DPN benchmark.

             The DPN method is equivalent to the response matrix/discrete ordinates method (RM/DOM) with double Gauss quadrature [4] for the exiting intensities in directions    [-1,0],[0,1] [see explanation of Eq(1)]. RM/DOM will also serve as the standard of comparison to verify the number of places of significance of the DPN benchmark.  Rather than discretize, we proceed analytically with an expansion of the angular flux in half-range Legendre polynomials in order to separate the forward and backward flow streams, which is entirely different from the benchmarks discussed in the Appendix.  We introduce the expansions into the transport equation, which, when projected over the half-range polynomials, yields two coupled first order vector equations for the moments of the expansion in the positive and negative directions.  With re-arrangement, the equations take on the parity form, decoupling one equation into a second order vector ODE, which we solve in the same way as for RM/DOM but now for the moments not fluxes.  The angular flux results as a truncated series expansion from the linear combination of the parities.  The numerical implementation then follows the RM/DOM benchmark.  The description of the PN method follows.

Added appendix

APPENDIX: Previously Published High- Precision Benchmark solutions

1. The response matrix discrete ordinates method (RM/DOM) [4]

The response matrix solution follows the Analytical Discrete Ordinates (ADO) method of Siewert [22], with one major exception.  While ADO provides a solution based on the discrete form of the singular eigenfunctions, RM/DOM assumes the even/odd parity form of the transport equation giving solutions of hyperbolic type to a second order ODE.  One advantage is that there always exists two independent solutions to the parity equations even in the case of zero absorption, which otherwise requires special attention.  In addition, the solution features the response matrix (as the name suggests) over a homogeneous slab of any thickness dependent upon material properties only.  Slab responses combine to construct the response of contiguous heterogeneous media under the interaction principle as described in [23].  Additional features include convergence acceleration in quadrature order through Wynn-epsilon acceleration and faux quadrature with quadrature grid refinement also enabling the beam source to be in any direction.  We have achieved seven-place precision for the Cloud C1 300-term scattering kernel [24] with quadrature order 2N of ~300.

2. Method of Doubling (MoD) [25]

This solution use the concepts developed by Van De Hulst [26].  The procedure is applied to the fully discretized 1D radiative transfer equation in both space and direction.  The response of a thin slice of a discrete slab is continuously doubled until the entire discrete slab is covered.  The full slab is then fully covered by adding via the interaction principle [23].  Richardson and Wynn-epsilon convergence acceleration apply to the initial thin slice and to the quadrature order respectively.  In addition, Richardson extrapolation is applied to the original slab discretization.  The features found in RM/DOM are also included.  Heterogeneous media are a naturally consequence of adding.  The Cloud C1 [24] case is found to be as precise as that of RM/DOM.  The advantage of doubling is that no matrix diagonalization with eigenvalues is necessary and essentially a host of spatial discretization schemes to evaluate the matrix exponential, as a solution to a first order ODE are possible.  Doubling demonstrates that the most basic of numerical finite difference schemes provides highly precise numerical solutions to the radiative transfer equation thus demonstrating its simplicity.

3. Matrix Riccati Equation method (MREM)[27]

The centerpiece of the Matrix Riccati Equation Method is the set of four non-linear matrix Riccati ODEs that represent the radiative transfer equation as a first order ODE similar to invariant embedding.  Any two of these equations gives the interaction coefficients of reflectance and transmittance for a slab.  These coefficients are found numerically from a TS representation to initiate doubling for a thin slab.  Forming the interaction principle with the interaction coefficients and doubling then enables the slab response for any given thickness.  With the response, the exiting and interior intensities are found.  Thus, we have established a numerical method based on TS evaluation, doubling and adding which we wrap in the Wynn-epsilon convergence acceleration to give a highly precise solution.  Tables for the HAZE L and Cloud C1 [24] benchmarks to nearly seven-place precision are calculated.  All exiting intensities are precise to seven digits in comparison to RM/DOM.

Note: All relevant reference included.

3. Could you summarize the proposed DPN approximation algorithm in steps?

Response:

For further clarity, the theoretical and numerical steps for the determination of the DPN benchmark are as follows:

1. Express the neutron angular flux of the transport equation as a series of orthogonal half-range Legendre polynomials and spatial moments for neutrons moving in positive and negative direction cosines (±m). [Eqs(2a,b)]
2. Since the scattering integral on the RHS of Eq(1a) is expressed as a full range moment, it is re-expressed as a sum of equivalent half-range moments. [Eq(5)]
3. The transport equation is then projected over half-range Legendre polynomials in the forward and backward directions to define two coupled first order ODEs for the spatial moments in each direction. [Eqs(7e)^±]
4. After application of the closure condition on the last moment (derivative set to zero), the ODEs are put in vector form and added and subtracted to form the parity equations. [Eqs(12a,b)]
5. By differentiation, the even parity equation becomes a single inhomogeneous second order ODE and the odd parity equation remains first order. [Eqs(18)]
6. The even parity equation solution is expressed as the sum of the solution to the homogenous and particular parts. Note that the solution to the homogeneous part is constructed form matrix functions with assumed (known) boundary conditions. [Eq(24c]
7. With the homogeneous solution known, the particular solution comes from variation of parameters. [Eq(25a)]
8. The exiting spatial flux moments are recovered by deriving the response matrix connecting the input moments to the output moments across the slab. [Eqs(34)]
9. The slab interior spatial moments then come from adding the parity components. (36a) ^±
10. With known spatial moments, the last step is the numerical evaluation of the half-range Legendre series using the Clenshaw algorithm and Wynn-epsilon convergence acceleration.

4. Check and revise Eq. 4

Done

5. To improve the overall quality of the paper, it is essential to well-written the manuscript including the English language as well as the punctuation mistakes.

Punction checked. English is also fine and much better than the above sentence you wrote.

6. Recent references must be added.

Done

Comments on the Quality of English Language

Editing of English language required.

Don't think so according to other reviewers and considering I have 320 publications This is an erronous statement which I challenge in the strongest way.

Reviewer 5 Report

Comments and Suggestions for Authors

See attached

Comments for author File: Comments.pdf

Author Response

2. Improved Introduction:
o Enhance the introduction by citing relevant works to inform readers of the main points, 
literature review, differences, and novelty.

Response : See revised1 version lines 27-68

1. PN Method Description:
o Before using the PN method, provide a detailed explanation of its long work history and 
fundamentals.

Response: See revised1 version lines 71-90

3.Modern References:
o
Cite modern references and revise the reference style according to the journal's guidelines.

Response: I have updated the references.4. Equation Definitions:
o
4. Clearly define Equations (1b).

Response: See lines 94,100-101

5.Clarify Symbols:
o
Explain the meaning of ± symbols used after some equations.

Response: + is equation with top symbol - is for equation with  bottom symbol (see lines 115-116) and line 402

6. Consistency in Symbols:
o
Ensure consistent use of ± throughout the paper.

Response: Seems OK

7. Equation Labeling:
o
Correct the labeling of Equation (11d), which should be revised from (11d ±.

Response: Got it.

8. Equation Check:
o
Verify the accuracy of Equation (14b).

Response: It's fine.

9. Punctuation in Equations:
o
Ensure proper punctuation (commas and periods) at the ends of equations.

Response: I always do.

10. Limitations and Motivation:
o
Discuss the limitations and motivation behind the work.

Response: Physical limitations in the title. Numerical limitations see lines 664-666

Motivation: See lines 6-7,39-49

11. Results Comparison:
o
Compare the results with previous works to highlight improvements and differences.

Response: That is what the tables are.

12. Visual Representation:
o
Add plots to illustrate the solutions.

Response: Do not need figures since I am only interested in number comparison. 

13.
Figure Labels:
o
Label the axes in Figure 1 clearly.

Response: Clarified with Eq(50)

14. Applications:
Discuss any potential applications of the work.

Response: Discussed benchmark comparisons. That is the application. Nothing need more be said.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The author replied to all my comments providing detailed explanations. Moreover, he improved the bibliography. 

Thus, I accept the paper in the current form.

Reviewer 4 Report

Comments and Suggestions for Authors

Thank you 

Reviewer 5 Report

Comments and Suggestions for Authors

The revised version is acceptable