Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge–Kutta Methods

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript provides an insight on how the combination between  adaptive time stepping and parallel-in-time methods have an impact on the tolerance proportionality. Grid refinement strategies are compared, the results show that a simple modification to the original refinement factor leads to a better computational stability and reliability. Computational experiments computational made using the XBraid library, prove that  parallel-in-time results  match those of sequential computations. The orhanization of the paper is as follows. Section 1 is introductory in nature. Section 2 contains preliminary issues and concepts needed in the sequel.  Section 3 approaches the tolerance proportionality and computational stability for the proposed parallel-in-time method with a modified step size selection strategy; numerical examples are provided related both to rodinary or partial dierential equations.  Section 4 presentssome open problems by highlighting some limitations of the proposed procedure. The abstract could be shorter (the first two sentences to not match there in my opinion). The results are interesting, and the numerical exmples are meaningful. The fugures are clear and well designed. The manuscript deserves publication.

Author Response

Our detailed answer can be found in the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper should have the following areas that need further improvement：

1 The algorithm description of adaptive parallel-in-time Runge–Kutta methods is unclear. 

2 The advantages and disadvantages of the adaptive parallel-in-time Runge–Kutta methods are not described in detail.

3 The framework structure of the paper is not good, it is suggested to divide the algorithm description section into one section.

4 Add numbers to the algorithm application examples and provide detailed descriptions one by one to increase readability.

5 The introduction section of the paper needs to include some latest literature research results and descriptions of the innovative points of the paper.

6More tables should be added to quantitatively analyze and compare the quality of algorithms. Too many graphics can be partially removed.

Author Response

Our detailed answer can be found in the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Suggest writing the algorithm on page 8 in the form of Algorithm 1.

After modifying the language, layout, and formatting of the paper, it can be published.