A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

Round 1

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 1.

We are grateful to the reviewer for the positive attitude towards our manuscript and also for the comments. We provide a point-by-point response as follows.

“1. Rephrase the sentence in abstract line 10 i.e. we extensively examined 13 numerical method…

in building wall.”

We have now changed this sentence to the following:

“Thus, in this work, we extensively examined 13 numerical methods to solve the linear heat conduction equation in building walls. Eight of the used methods are recently invented explicit algorithms which are unconditionally stable.”

 

“2. Rewrite all the double citation in the format e.g. [5,6] instead of [5],[6].”

Actually, this format is due to the reference manager Mendeley and currently, we don’t know how to change it. However, during the proofreading, these issues are going to be fixed, anyway.

 

“3. Correct N in line 130.”

It has now been corrected, thanks to the reviewer. The reason of this problem was that the pdf maker couldn’t handle the cross-product sign of the MathType properly.

 

“4. Rewrite the number 0.0234.0.98^99 in line 145”

It is fixed now, thanks to the reviewer.

 

“5. Define C in Eq (15)”

We wrote after that equation: “where C is the heat capacity defined in Subsections 2.1 and 2.2.” and indeed, we explained it in more detail in those subsections.

 

“6. Write a proper methodology for at least one method i.e. how the Eq of DF algorithm in

line 261 are obtained, or Eq (12) and (13).”

We have now inserted several lines to explain how the formulas can be obtained.

 

“7. To show the better understanding of the effects observed in the Figures and Tables, the

authors can explain some additional notes about these.”

We have now inserted some additional explanations around Figures 8 and 15.

“8. The author should improve the introduction section by adding the following recent work

in the literature.

https://doi.org/10.1007/s00366-021-01327-5

https://doi.org/10.1007/s00366-019-00760-x

10.1002/adts.202100600”

We have now added all the recommended citations, thanks to the reviewer.

 

“9. Generally, English and presentations are acceptable, but some English and statements should be clarified and improved for publications.”

We have now gone through the text again and fixed some grammar and style issues, mostly following the recommendations of the other reviewer. 

Reviewer 2 Report

Dear authors,

apart from the comments provided below, please also see the annotated manuscript for further comments (especially regarding spelling and writing suggestions).

The article deals with the analysis and comparison of different time stepping methods for the heat conduction equation. In that regard, several unconditionally stable explicit methods are employed and investigated in terms of their accuracy. Despite the fact that the article is generally well-written, there are some issues that must be addressed by the authors in the revision.

1) Title: The title should be changed since no general numerical methods are investigated (which I would associate more with the spatial discretization), but time integration schemes are studied. This should be reflected in the title. Replace "numerical methods" by "time integration schemes"

2) Improve the resolution of all figures. Only high-res pixel graphics or vector-graphics should be employed.

3) More details regarding the spatial discretization technique, which is finite difference like, must be added.

4) In line 111, a thermal resistance is introduced, which is neither part of Eq. (1) nor (2). Therefore, their use in Eq. (3) is not explained to the reader. Please provide a more comprehensive discussion.

5) The authors use h to denote the time step size. This is an unfortunate choice, since typically h refers to the mesh size and Δt is used for the time increment. Maybe, it is a good idea to stick to that convention.

6) In the analysis of the different time stepping methods, only the accuracy is addressed. This is not enough since the computational costs are a second important factor. It is of not much use if a method is very accurate for a given mesh and time step, but the simulation takes order of magnitude longer. Therefore, the authors must include also this aspect in the discussions.

The authors must address all of the issues mentioned above as well as provided in the annotated manuscript in their revision.

 

Kind regards.

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 2.

Dear Reviewer, we provide a point-by-point response as follows.

 

“apart from the comments provided below, please also see the annotated manuscript for further comments (especially regarding spelling and writing suggestions).”

We are really grateful for sacrificing your precious time to help us improve our manuscript. We have gone through carefully on the attached version of the manuscript and performed all the required changes.

“1) Title: The title should be changed since no general numerical methods are investigated (which I would associate more with the spatial discretization), but time integration schemes are studied. This should be reflected in the title. Replace "numerical methods" by "time integration schemes"”

We now have changed the title.

 

“2) Improve the resolution of all figures. Only high-res pixel graphics or vector-graphics should be employed.”

There were complaints in the case of our previous papers on the quality of the figures, made by saving the figures of MATLAB in jpeg format. Now we have found how to increase the resolution and reproduced all of these figures.

 

“3) More details regarding the spatial discretization technique, which is finite difference like, must be added.” and

“4) In line 111, a thermal resistance is introduced, which is neither part of Eq. (1) nor (2). Therefore, their use in Eq. (3) is not explained to the reader. Please provide a more comprehensive discussion.”

We now have added almost a full page about spatial discretization, including the definition of thermal resistance. We admit that previously the information about this was a little scarce in our manuscript.

 

“5) The authors use h to denote the time step size. This is an unfortunate choice, since typically h refers to the mesh size and Δt is used for the time increment. Maybe, it is a good idea to stick to that convention.”

We now have changed h everywhere to Δt. We note that previously we had used h because we applied the logic of the method of lines where the spatial discretization is already done and then one uses ODE solvers.

 

“6) In the analysis of the different time stepping methods, only the accuracy is addressed. This is not enough since the computational costs are a second important factor. It is of not much use if a method is very accurate for a given mesh and time step, but the simulation takes order of magnitude longer. Therefore, the authors must include also this aspect in the discussions.”

The reviewer is completely right. We have now produced two figures where the running times are on the horizontal axis. Actually, since all of these are fully explicit methods, the running times are roughly proportional to the number of stages.

 

“The reviewer, is not familiar with the previous works, but explicit and unconditional stability of a time integrator never go hand in hand. Maybe, there is a misunderstanding in the terminology and what the authors try to express.”

In the introduction, we have now given the (usual) definition of stability in an extra sentence: “Unconditional stability here means that the temperature remains finite (i.e. errors are not amplified without bounds) for arbitrary time step size.” Actually, the Dufort-Frankel method as an explicit and unconditionally stable method for the heat equation is mentioned by textbooks as well, but they don’t say it cannot be unstable for other equations.

 

“In the multilayer case, always the left 50% of the cells were brick and the right 50% were insulator. - That is weird? Why is that assumption necessary? Isn't it enough to work with 2 models?”

Actually, this was the simplest and most convenient way to implement different non-uniform meshes and different materials at the same time. If we would fix the width of the brick and the insulator, then we couldn’t be able to apply so simple rules for the cell-sizes as explained in Subsection 2.2 or, the border of the material would be inside the cell.

 

“What are the particular advantages and disadvantages of such unconditionally stable explicit algorithms? Does it ever make sense to employ conditionally stable explicit algorithms in heat problems?”

We have now inserted the explanation as follows:

“Actually, the price of unconditional stability is conditional consistency, which means that spatial mesh refinement with a constant time step size yields worsening accuracy (in contrast to worsening stability properties as in mainstream methods), which is examined analytically and numerically in our previous papers [44] and [35], respectively. Below in Fig 9, we will show an example when the conditionally stable Heun’s method is significantly more accurate for small time step sizes than our methods.”

Round 2

Reviewer 2 Report

Thank you for addressing the issues mentioned in the review.