Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness
Round 1
Reviewer 1 Report
The paper is of good scientific quality that is well written and the material presented will be of interest to the scientific community at large. The authors employ an explicit finite difference scheme in combination with the Monte Carlo approach to solve a class of random reaction-diffusion equations.
However, there are a few questions/concerns that need to be addressed:
How does the proposed approach compare to alternative approaches, in particular, how would a different numerical scheme instead of (15) behave in comparison? Why was the standard scheme (15) chosen; any particular advantages? Conditions are provided for the positivity of the numerical solution, but how do those conditions compare if another classical discretization scheme was used?
What is the motivation for this particular class of reaction-diffusion equations (3)-(6) with conditions (7)-11) – two references are provided, but the paper would benefit if a more specific explanation is given?
The paper would also benefit from a Conclusion section, where the most important contributions of the paper are summarized.
In addition, the paper would benefit from a few more references related to alternative numerical approaches for solving this and similar types of problems.
Also, the font size in Figures 2 (a) and (b) could be bigger- for clarity and consistency with Figure 1.
The English grammar/spelling is satisfactory, however there are several minor needed changes. The second half of the Abstract can be rewritten for clarity of the presentation, as well as some parts of the main presentation. There are missing/misplaced commas in several sentences, some mixture of present and past tense that should be avoided, and other minor grammatical issues. To name a few:
Line 6: replace “throughout” with “through”
Line 18: replace “instead” with “instead of”
Line 46: remove the extra space at the beginning
Line 52: move the sentence to the previous paragraph, i.e., Line 51.
…
Author Response
Answers to Reviewer #1.
- How does the proposed approach compare to alternative approaches, in particular, how would a different numerical scheme instead of (15) behave in comparison? Why was the standard scheme (15) chosen; any particular advantages? Conditions are provided for the positivity of the numerical solution, but how do those conditions compare if another classical discretization scheme was used?
Answer: As it is written in the manuscript introduction, see pages 1-2, lines 27-32, implicit iterative methods are unsuitable because they do not allow the computation of the statistical moments of the approximate s.p. solution. The scheme (15) is chosen because it is explicit, efficient and easy to implement. Since we try to use Monte Carlo method to avoid the computation breakdown due to the storage accumulation, the proposed method is particularly advisable.
Other possible schemes under some additional conditions, such as Crank-Nicolson could be tried but with a particular ad hoc analysis of the positivity and stability.
All these comments are included in the Conclusion section 5.
- What is the motivation for this particular class of reaction-diffusion equations (3)-(6) with conditions (7)-(11) – two references are provided, but the paper would benefit if a more specific explanation is given?
Answer: The problem is widely used in the deterministic framework, here we take into account uncertainty in the model. Conditions (7)-(11) are accessible conditions that will be required later in the proofs of the results and that are convenient to state clearly the problem. We have added a more specific explanation in page 3, lines 55-57, and several additional references have been included, see [17],[18],[19] and [20].
- The paper would also benefit from a Conclusion section, where the most important contributions of the paper are summarized.
Answer: We have added a Conclusion section.
- In addition, the paper would benefit from a few more references related to alternative numerical approaches for solving this and similar types of problems.
Answer: We have added three more references [7], [8] and [9], related to your query and a comment in the Introduction section, page 1, lines 24-27.
- Also, the font size in Figures 2 (a) and (b) could be bigger- for clarity and consistency with Figure 1.
Answer: Font size in Figures 2 (a) and (b) has been increased.
- The English grammar/spelling is satisfactory, however there are several minor needed changes. The second half of the Abstract can be rewritten for clarity of the presentation, as well as some parts of the main presentation. There are missing/misplaced commas in several sentences, some mixture of present and past tense that should be avoided, and other minor grammatical issues.
Answer: Grammar typos have been corrected.
Reviewer 2 Report
Referee Report for the Paper
This manuscript provides the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion-reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. Results are simulated and a procedure for the numerical computation is given.
The idea of the paper is interesting; to the best of my knowledge. The manuscript is clear and substantially easy to read.
There are some comments the author could deal with, or at least discuss.
1- The Algorithm section of the manuscript was not written in the high-quality typesetting system; please modify it.
2- The number ''e'', also known as Euler's number should be written with this Command ''\mathrm{e}''.
3- A conclusion is also needed to illustrate your main results.
My overall evaluation of the paper is positive, provided that the above improvements are made.
Author Response
Answers to Reviewer #2.
- The Algorithm section of the manuscript was not written in the high-quality typesetting system; please modify it.
Answer: Algorithm writing has been typeset properly.
- The number ''e'', also known as Euler's number should be written with this Command ''\mathrm{e}''.
Answer: Euler’s number has been well written.
- A conclusion is also needed to illustrate your main results.
Answer: We have added a Conclusion section.
Round 2
Reviewer 1 Report
The paper is acceptable for publication in its current (revised) form.
Author Response
We thanks again the comments and suggestions of the reviewer.
Reviewer 2 Report
There are some typos in the manuscript:
In the Introduction section, Lines 24 and 25, the verb “is” does not agree with the subject, and the word “in appropriate” seems to be miswritten; please correct them.
Author Response
Typos have been corrected. Thank you very much
Round 3
Reviewer 2 Report
My overall evaluation of the paper is positive.
