Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

Round 1

Reviewer 1 Report

My comments:

Please avoid bring citations in the part abstract since sometimes somewhere abstract will be appeared without list of references.

Please make the abstract shorter.

Please remove the works in the long list of bibliography if you did not use them directly.

I will review the manuscript scientifically later if the quality of presentation of it gets improved.

Author Response

The reviewer is correct when saying we shouldn’t insert references into the abstract. So, thanks to the reviewer, these are now deleted and the abstract is made significantly shorter. We also deleted 12 items from the list of bibliography which we thought to be the less important ones, thus the introduction became also shorter. We went through the manuscript thoroughly and made many small changes to improve the quality of the presentation.

Reviewer 2 Report

The present paper can be viewed as an interesting addition to a series of papers on numerical algorithms to solve the heat or diffusion equation. The new results are interesting and can be applied to real problems.  

The results appear scientifically sound and they complement and enrich the results of recent papers in the literature. However, some points need to be revised according to the following list of comments.

For this reason I recommend a minor revision of the paper.

I suggest a revision of the English throughout the paper. Many typos need to be corrected and some sentences need to be written more clearly. In particular, Section 4 needs to be better arranged. Just as an example

• Line 265: “The computer program has…..”

and so on…

• Lines 555-556 could be substituted by: The present paper is the natural continuation of previous studies on explicit algorithms for solving the time-dependent diffusion equation.  
• Line 582-583 could be substituted by: Our next step is to search more accurate algorithms for nonlinear parabolic equations.
• Lines 586-589 could be substituted by: These results are of great interest to simulate real problems such as, in engineering, the heat transfer by conduction and radiation in elements of machine-tools.

Author Response

Thank you for the positive attitude towards our manuscript and the benevolent suggestions. We performed all the proposed substitutions, then went through the manuscript thoroughly, corrected some typos, and made some other, small changes to improve the manuscript.

Round 2

Reviewer 1 Report

My comments have been considered completely.