A Hybrid Time Integration Scheme for the Discontinuous Galerkin Discretizations of Convection-Dominated Problems

Round 1

Reviewer 1 Report

A Hybrid Time Integration Scheme for the Discontinuous Galerkin Discretizations of Convection-Dominated Problems.

In this paper, authors develop a hybrid time integration scheme based on the idea of Kanesvsky et al. (2007) of combining an explicit and an implicit scheme. The explicit scheme is a strong stability preserving Runge-Kutta (SSP-RK3) scheme well known. The implicit method is also well known, they use a diagonally implicit Runge – Kutta (Alexander 1977). To ilustrate the performance of the proposed hybrid method, the authors conduct several numerical tests.

The article is well written in general but I think it doesn't fit with the subject Areas of "Energies".

The abstract is difficult to read (the first sentence occupies 5 lines) and it is not clear the objective of the article or the main result of the study.

Concerning the Introduction, there are several reviews written on the subject that should be cited. For example, Gottlieb, Sigal, Chi-Wang Shu, and Eitan Tadmor. "Strong stability-preserving high-order time discretization methods." SIAM review 43.1 (2001): 89-112.

The authors test the hybrid time integration scheme they propose using 5 benchmark examples, 3 in 1D and 2 in 2D. It is not clear to me what the purpose was to present each problem and I would have preferred to find only two examples in greater detail about consistency, convergence, stability and performance with respect to other schemes.

A comparison / discussion with other methods of the literature for the study of convection-diffusion problems is also needed in section 5. See for instance (H. Egger, J. Schöberl [2010] A hybrid mixed discontinuous Galerkin finite-element method for convection–diffusion problems, IMA Journal of Numerical Analysis)

In my opinion, to be published, the article should answer the questions: why is this hybrid scheme useful for realistic applications? And what is the novelty compared to other existing schemes?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper (energies-744150) under review deals with the research on the hybrid scheme comprising the 3-stage, 3rd-order accurate, strong stability preserving Runge-Kutta (SSP-RK3) scheme and the 3-stage, 3rd-order accurate, L-stable, diagonally implicit Runge-Kutta (LDIRK3) scheme for the temporal integration of the discontinuous Galerkin (DG) discretization for the time-dependent convection-dominated problem. The proposed model develops the existing DG method. The paper tackles an important issue in the heat and mass transfer and fluid dynamics, and therefore is suitable for Energies. The experimental part of the article contains crucial information regarding the research model, method, numerical test and technique. A structure of the paper is in accordance with principles of good scientific reports.

Comments: in opinion of the reviewer the abstract is somewhat confusing.

The paper can be accepted for publication.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Brief Description of the Work

The discontinuous Galerkin method is a type of finite-element scheme in which the solution across cell boundaries can be discontinuous. The method was originally conceived for solving elliptic equations (Nitsche - 1971), and successively updated for solving hyperbolic PDEs (Reed and Hill - 1973). Later on, the method has been extended for solving also nonlinear systems of equations (Cockburn and Shu -1998).

In this work, the authors propose a hybrid scheme for the time-dependent integration of the DG discretization related to a two-dimensional convection-diffusion problem (see PDEs (1) at page 2 of the manuscript). The hybrid scheme, developed by the authors, consists of the three-stage Strong-Stability Preserving Runge-Kutta (SSP-RK3) scheme and the three-stage L-stable, and diagonally implicit Runge–Kutta (LDIRK3) scheme. By separately treating the coarse elements and the refined elements with an explicit scheme and an implicit scheme, the authors showed that it is possible to get a time step bigger than that for a purely explicit scheme at a comparatively low cost.

Questions/Suggestions

 Q1) Region of absolute stability.

The authors proposed a strategy of grouping the elements able to maintain the stability and to reduce the computing cost of the hybrid scheme. In this respect, benchmarking with other reference results have also been performed by them. However, as is customary in these cases, I would have expected to see a plot illustrating the region of absolute stability for their hybrid scheme compared to the (traditional) three-stage SSP-RK3 and LDI-RK3 schemes. In addition, what about the absolute stability when there is no diffusion in the system ? May the authors add this (or these) plot(s) ?

Q2) Mesh quality.

The authors explained in an accurate and detailed way the "time integration" and "grouping and scheme coupling" aspects related to their hybrid numerical scheme. However, I did not notice the same degree of accuracy when they dealt with the "DG discretization" topic. More specifically, as well-known, the type (in particular the stability) of the mesh can be decided on the base of three main indicators: skewness, aspect ratio and smoothness. In this regard, the authors are kindly invited

Q2a) To specify which kind of methods they have used for determining the skewness of their grid (e.g. "equilateral volume", "deviation from normalized equilateral angle", "equiangular skew");

Q2b) To specify if the aspect ratio is equal to 1 - or, at least, near to 1 - and that adjacent cell sizes do not vary by more than 20% (to avoid big interpolation error);

Q2c) To show that in their scheme there are not sudden jumps in the size of the cell (to avoid of getting wrong results at nearby nodes).

Q3) Three-stage Runge-Kutta (SSP and LDI.

The authors studied the case of three-stage Runge-Kutta SSP and LDI. As known, for solving the following P.D.E.

F(f,t)=f+Δt L(f,t)

where F indicates a first-order Euler is the L is RHS operator from the spatial discretization of the DG scheme, we may also use the four-stage SSP-RK3 allowing twice the CFL (for the cost of additional memory) with respect to the other three-stage schemes. The main result is that for the four-stage SSP-RK3 scheme the CFL ≤ 2. However, it allows twice the time-step than the three-stage SSP-RK3, resulting in a speed up of 1.5? compared to the three-stage SSP-RK3 scheme. The question is: did the authors compare the hybrid method that they developed with the four-stage SSP-RK3 in terms of stability and computing cost ?

Conclusions

The work is certainly interesting and the idea proposed by the authors is original. However, as mentioned above, it shows aspects of vulnerability and incompleteness that, in my opinion, should be filled. Written in this form, the article may be subject to criticisms form the scientific community working in the field. However, I believe that if the authors are disposal to make an additional effort, responding adequately to the points raised in Q1), Q2) and Q3) above, their work will acquire more soundness and it will be appreciated by experts working in the field.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

According to the revision, most of my concerns have been clarified. I think that the article has improved substantially.

Reviewer 3 Report

The authors answered in a satisfactory way to all the questions raised in my previous report. The supplementary documents, added by the authors in this second round of evaluation, show that the work has been accomplished with professionalism. If possible, it would be very nice if the authors may publish not only the article but also the annexed documents.
I am convinced that the work will be appreciated by specialists in the field.