On the Selection of Weights for Difference Schemes to Approximate Systems of Differential Equations
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this submission, the authors tried to investigate how the weights compute be computed for constructing a higher order s-stage method under the family of RK family of methods. This could be viewed as extensions over the explicit conservative Runge-Kutta schemes.
The values of the number of stages s and the Butcher coefficients are known, which provide approximation up to the 14th order based on trees.
Some comments are in order:
Under a polynomial right-hand side and another simplification, the new weights have been discussed.
Why in Figure 2-right, the label says it is a fourth order scheme while the slope is roughly two?
Apply the weights in Table 3 to solve an ODE problem and compare the slopes of the numerical rates with other solvers such as RK4.
The motivation and the goal of the submission have not been treated in Section 1. This is important since it is not clear why we need such higher order time-stepping solvers. Mainly in practice, RK solvers up to 4th order or Krylov-type methods are used to solve large-scale ODE systems.
The language of the paper is not smooth.
Comments on the Quality of English LanguagePlease see above.
Author Response
First of all, we want to thank the reviewer for the detailed feedback
Comments 1. In this submission, the authors tried to investigate how the weights compute be computed for constructing a higher order s-stage method under the family of RK family of methods. This could be viewed as extensions over the explicit conservative Runge-Kutta schemes.
The values of the number of stages s and the Butcher coefficients are known, which provide approximation up to the 14th order based on trees.
Some comments are in order:
Under a polynomial right-hand side and another simplification, the new weights have been discussed.
Response 1: We considered a broader problem: the selection of weights for difference schemes described by symbolic expressions. We used Runge-Kutta schemes to test our approach. These schemes are convenient because they are very well studied. We added a clarification to the Introduction, it is highlighted in red
Comments 2. Why in Figure 2-right, the label says it is a fourth order scheme while the slope is roughly two?
Response 2: The order of Shanks scheme is four only for linear and scalar problems. It is a counterexample for the hypothesis that scalarization always gives the correct weights. For clarity, we have added an example at the end of the article
Comments 3. Apply the weights in Table 3 to solve an ODE problem and compare the slopes of the numerical rates with other solvers such as RK4.
Response 3: we added at the end of the article an example with Roessler system.
Comments 4. The motivation and the goal of the submission have not been treated in Section 1. This is important since it is not clear why we need such higher order time-stepping solvers. Mainly in practice, RK solvers up to 4th order or Krylov-type methods are used to solve large-scale ODE systems.
Response 4: We added a clarification to the Introduction, it is highlighted in red.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper deals with an important task. It has a logical structure. The paper is technically sound. The proposed approach is logical. My suggestion is to recommend the manuscript for publication with major revisions:
1-The part of numerical is poor. It should give more examples with different cases. My opinion is It will be good if the authors use applications such as physical or biological nonlinear mathematical models with real data and solve it with the suggested method. I think the results will be very interesting.
2-The authors must add analysis for the suggested method such as the convergence and the stability with error analysis.
3 They should double-check the mathematical formulations, and add appropriate references for governing equations.
4. Make sure all cited references are correct.
5. A professional proofreading revision is required.
Comments on the Quality of English LanguageA professional proofreading revision is required.
Author Response
First of all, we want to thank the reviewer for the feedback.
Comments 1. The part of numerical is poor. It should give more examples with different cases. My opinion is It will be good if the authors use applications such as physical or biological nonlinear mathematical models with real data and solve it with the suggested method. I think the results will be very interesting.
Response 1: We have added on page 8 an additional example with the Roessler system. This example illustrates the difference we found in the solution of linear and nonlinear systems within our approach.
Comments 2. The authors must add analysis for the suggested method such as the convergence and the stability with error analysis.
Response 2: We use Richardson's method to estimate errors. This method is built into our fdm system and is described in [5]. We have added the necessary comments on pages 6 and 7.
Comments 3. They should double-check the mathematical formulations, and add appropriate references for governing equations.
Response 3: We've added some references.
Comments 4. Make sure all cited references are correct.
Response 4: We re-checked the references
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThis version is fine.
Reviewer 2 Report
Comments and Suggestions for AuthorsI am satisfied with the changes made. Therefore, my suggestion is to recommend the manuscript for publication.
Comments on the Quality of English LanguageA professional proofreading revision is required
