A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This work is concerned with extending a class of numerical method for conservative hyperbolic systems to the non-conservative case. There are many physical systems that can be cast in such non-conservative form, such as some moment closure models of rarefied transport. However,  numerical methods for such non-conservative systems is limited. The presented work aims at bridging that gap.

In general the paper is well written and easy to follow, however i do have some major issues that requires clarification:

1. The authors start with a formulation in (8) which would only make sense if the test functions are piecewise constant in time. In a Galerkin method (not a Petrov-Galerkin method) one uses the same spaces for test and trial, and so i was under the impression that the trial functions are also piecewise constant in time. However, later on the in the paper, the authors introduce a higher order discretization in the time variable (see equation 15).  I would like to ask the authors to clarify discretization spaces.

2. I also think its important to show consistency of the method. With consistency I mean tat the strong form are also solution to the weak and discrete forms? Without showing consistency, its unclear how solutions of the discrete equations relate to solutions of the strong equation.

3. I think its also important to show stability of the method in a certain sense. A typical example would be to show entropy/L2 stability. However, stability in stronger norms is also desirable. Without stability guarantees, its hard to understand the robustness properties of the presented method.

Comments on the Quality of English Language

Seems OK to me.

Author Response

We sincerely thank the Reviewer for her/his suggestions that help us to improve the quality of our work.  We have carefully revised the article according to your comments. Please refer to the attachment for specific responses.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In the article the numerical method to solve the shallow water equations in non-conservative form is described and tested. The method is based on a weak formulation obtained within the framework of the discontinuous Galerkin method approach. It uses a polynomial representation of the solution in space and time for integration over time in the style of the ADER method. A number of tests with smooth and discontinuous solutions are presented.

Remarks on the article content.

- The use of the Taylor expansion basis polynomials for polynomial solution reconstruction is known to lead to the scheme poor stability because this basis polynomials turn out to be close to linearly dependent while rising polynomial degree. Is it possible to use some orthogonal polynomial system? Will it affect the method features?

- The presentation of the test calculations results lacks a description and discussion of the results. What does the tests analysis show? What does comparison of results with other works give? Thus, the results in section 3.2.2 differ from those presented in the references, especially at time t = 0.48. What does this difference mean?

Notes on design and typos.

line 18 - why exactly does the stream look like this (two components?)

line 27 - what are the matrix sizes?

line 36 - should be definition, not defination.

line 75 - it's not well to link forward to formula (6).

line 94 - the application of the designation from 1d framework in general case is not very clear.

line 119 - should be notations, not nations.

The tables 2, 3, 4 should be designed in style of table 5, especially in floating point numbers presenting.

Comments on the Quality of English Language

Moderate editing of English language required

Author Response

We sincerely thank the Reviewer for her/his suggestions that help us to improve the quality of our work.  We have carefully revised the article according to your comments. Please refer to the attachment for specific responses.

 

 

Author Response File: Author Response.pdf