An Efficient Finite-Difference Stencil with High-Order Temporal Accuracy for Scalar Wave Modeling

Round 1

Reviewer 1 Report

The paper presents the new stencil scheme for simulation of 2D constant density scalar wave equation. The paper is a valuable scientific text, since it includes all relevant references with appropriate review, explains stencil construction method in detail, provides the coefficient derivation details, illustrates the new method with simulation benchmarks and stability and dispersion analysis. In total, I consider that the paper has everything necessary for the introduction of the new stencil for seismic modelling.

Thus, I believe that the paper should be published. 

 

I would like to specially note the valuable study of compute time comparison with different methods, shown in Table 2. 

 

However, I would like to recommend the authors to focus their following works on more relevant mathematical models, such as 3D wave equation, systems of equations of elasticity with anisotropic models, because I believe that this is more in demand in the recent computation software development. Since the anistropic model equations are hardly possible without rotated cross stencil, such simulations can take advantage of the new stencil proposed in the current paper.

 

Here are some other minor comments to the text:

 

1. If derivation (1)->(2) follows [Dablain, M. The application of high-order differencing to the scalar wave equation. Geophysics 1986, 51, 54–66.], please mention, as it was in the original text, that it is possible only for constant v.

2. Please increase the legend font size in Figure 2

3. Why do you choose M=N=8 in the benchmarks? Personally, I would be interested in variants with N=M/2. How do you recommend the reader to decide on the best M, N values for their problems?

4. Please elaborate on the stability factor (22): its theoretical definition and derivation for the current paper

5. What are a_n, b_n coefficients equal to in the benchmark? Even though the reader can compute with the use of expression (12), a table with numerical values for some cases of M, N can be interesting to compare the method with others.

6. Please provide the formula expression for the relative error computed in section 5.2

7. The performance for memory bound problems can be estimated by the ratio of the number of values loaded per stencil to the memory throughput of the computer. I believe that the study of the theoretical performance limits would highly benefit the authors findings (see also RoofLine model)

 

Still, I believe that the study has scientific interest and is well written, thus, it should be published. The application of the proposed method to more complex schemes may be executed by an interested reader. 

 

 

Author Response

Reviewer 1:

Thanks for the reviewer's valuable advice, especially the suggestion that we extend our work to 3D wave equation, and the systems of anisotropic models. We made revision according to reviewer’s suggestions, and replied some important comments as follows:

1. If derivation (1)->(2) follows [Dablain, M. The application of high-order differencing to the scalar wave equation. Geophysics 1986, 51, 54–66.], please mention, as it was in the original text, that it is possible only for constant v.

Thanks for the reviewer's comments, we have modified this problem.

 

2. Please increase the legend font size in Figure 2

Thanks for the reviewer's comments, we have modified this problem.

 

3. Why do you choose M=N=8 in the benchmarks? Personally, I would be interested in variants with N=M/2. How do you recommend the reader to decide on the best M, N values for their problems?

We choose the higher-order example of M=N=8, because as the order increases, the difference in accuracy becomes smaller and smaller. However, the computational efficiency of the new method is much higher than that of the conventional method in the higher-order case. So we choose this parameter to do the experiment, and perhaps there are also personal habits.

How to choose the optimal parameters is a worth problem. Dispersion error is related to the minimum and maximum values of velocity model, grid spacing, time step and M, N of the FD scheme. As it happens, our previous work”Chen, G.; Peng, Z.; Li, Y. A framework for automatically choosing the optimal parameters of finite-difference scheme in the acoustic wave modeling. Computers & Geosciences 2022, 159, 104948.” (reference [23]) has studied the strategy of selecting the optimal parameters, and this selection framework is also applicable to the proposed FD scheme proposed.

 

4. Please elaborate on the stability factor (22): its theoretical definition and derivation for the current paper

Thanks for the reviewer's comments, we have made a detailed supplement to equation (22).

 

5. What are a_n, b_n coefficients equal to in the benchmark? Even though the reader can compute with the use of expression (12), a table with numerical values for some cases of M, N can be interesting to compare the method with others.

Thanks very much for the reviewer's suggestion, we have added some cases about this equation( Lines 82-93 in new revision). The exact solution is related to the Courant number r, so we did not give the specific value. But with our addition to equation (12), the solution becomes easier to understand.

 

6. Please provide the formula expression for the relative error computed in section 5

Thanks for the reviewer's comments, we have modified this problem.

 

7. The performance for memory bound problems can be estimated by the ratio of the number of values loaded per stencil to the memory throughput of the computer. I believe that the study of the theoretical performance limits would highly benefit the authors findings (see also RoofLine model)

Thanks for the reviewer's suggestions. We will carefully investigate the theoretical performance limits and RoofLine model you mentioned.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Works that use finite differences with "cross stencil" or "(pi/4)-degree rotated cross stencil (a little less)" are known in the open bibliography, mainly to solve problems of heat and mass transfer and electric fields. However, the work analyzed here proposes to solve an equation (two-dimensional wave equation), which is not one of the most complexes in fluid mechanics, however, it uses the idea of the "radiation stencil" to solve it, which makes the work somewhat different from those in the open bibliography.

The authors present an introduction with a robust and pertinent bibliographic review for the focus of the work. With regard to sections 2, 3, and 4, they are well written, only needing a review of small errors such as putting the unit of measurement glued to the number, as for example occurs in line 118 (h = 4m) or in Figure 5 (Lateral (m)). It is important to note that this small typography occurs in a large part of the text.

In addition, I suggest that the authors detail the formulations step by step, such as equation (6) is generated by replacing equation (5) in equation (4). Since it is a work that uses a well-known equation, but with a method that is proposed to be new, it is important to detail the formulations so that they serve as a benchmark for new researchers.

I am curious to see this method applied to a system of non-linear differential equations and discover the numerical efficiency and especially the computation speed, because, for example, the errors presented (numerical precision) in Table 2 are reasonable, but the computation is very interesting.

I conclude by understanding that the potential of the work is good when we look at the proposal of the proposed numerical method and not the physical problem solved (which is simple).

Author Response

Reviewer 2:

We sincerely thank the reviewer 2 for their positive comments and valuable suggestions. We made revision according to reviewer’s suggestions, and replied some important comments as follows:

1. Works that use finite differences with "cross stencil" or "(pi/4)-degree rotated cross stencil (a little less)" are known in the open bibliography, mainly to solve problems of heat and mass transfer and electric fields. However, the work analyzed here proposes to solve an equation (two-dimensional wave equation), which is not one of the most complexes in fluid mechanics, however, it uses the idea of the "radiation stencil" to solve it, which makes the work somewhat different from those in the open bibliography

Thanks to reviewer for the positive comments. The open literature has similar radiation stencil, but I think they belong to different systems. Although they look similar, their motivitions and starting points are completely different, and solving their FD coefficients are also vastly different. Our motivation is to obtain a high-order approximation in time and space with good computational efficiency. Therefore, we study the classical temporal high-order stencil in detail, and propose the improvement strategy in this manuscript. 

2. The authors present an introduction with a robust and pertinent bibliographic review for the focus of the work. With regard to sections 2, 3, and 4, they are well written, only needing a review of small errors such as putting the unit of measurement glued to the number, as for example occurs in line 118 (h = 4m) or in Figure 5 (Lateral (m)). It is important to note that this small typography occurs in a large part of the text.

Thanks for the reviewer's comments, we have modified this problem.

 

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Good paper

Comments for author File: Comments.pdf

Author Response

We sincerely thank the reviewer 3 for their positive comments.