One can always create a second order discretization method, whose accuracy surpasses even a twentieth order method! Hence the question needs rephrasing, in terms of accuracy and not on the basis of order of method!! This obsession with order of method leads one nowhere. For a change, compare resolution in spectral plane. Thankfully, we have been doing this for quite sometime!! This is not the so-called Fourier analysis of a derivative etc. One should look at the accuracy of space-time dependent solution of generic canonical problems.

First, consider "high order method" and "highER order method" are different in CFD literature: High order method means anything second order of convergence. But higher order method means 3rd or higher order (like ENO or WENO schemes, compact difference method, spectral methods, spectral element, ....)

Back to your question:

The answer to this is problem dependent! for example spectral methods (it is not spectral element, I mean methods like Chebyshev polynomial) are converge very fast (in theory with exponential order of convergence) however, it is very hard or impossible to use them without special tricks on not-clean-cut geometries. Also the answer depends on your computational power and the accuracy you need. If you deal with a very large domain or repeated simulations (like Monte Carlo Simulations for design problems), then high order (or higher order) methods can save you a lot of computational resources. Also consider that implementing boundaries, special geometry features, and .... are not very straightforward in high order (higher order) codes.

If you do not develop an in-house code and you only use a CFD package for sure high order is better! you can reduce your mesh size but it is always a trade off between (larger mesh and low order method vs., smaller mesh and high order method) in practice you find your answer with trail and error when you you implement both and do an error study

https://hal.inria.fr/hal-01444075/document

First of all, many of these methods are related to each other, if not in principle, at least in the way in which they are implemented. For example, nodal DG methods are often implemented as spectral element methods. Then, as always with numerical methods, it is hard to say which method is 'best', since this depends a lot on the specific application. In general, methods based on the weak form of the equations such as DG can handle in an easier way complex geometries with respect to high order finite difference approaches. It is also quite widely acknowledged now that for orders higher than 2-3 finite volume approaches are more complicated to implement and involve more communication on parallel machines than DG methods. Personally, I believe that modal DG is attractive for adaptive CFD simulations, but on the other hand most of the recent developments on the analysis of DG methods have taken place for nodal DG approaches. A nice overview of recent developments in this area is given by the plenary talks at the ICOSAHOM conference held this year in London, which are available online at

http://icosahom2018.org/

One can always create a second order discretization method, whose accuracy surpasses even a twentieth order method! Hence the question needs rephrasing, in terms of accuracy and not on the basis of order of method!! This obsession with order of method leads one nowhere. For a change, compare resolution in spectral plane. Thankfully, we have been doing this for quite sometime!! This is not the so-called Fourier analysis of a derivative etc. One should look at the accuracy of space-time dependent solution of generic canonical problems.

Professor Tapan K. Sengupta you are right! However, the community is incredible interested on that issue even more than the true physics. That has lead a sour taste after attending recent conferences. I have seen a bunch of non-sense results but with very high order schemes and beautiful colors. The sad true is that funding agencies are pumping a bunch of money on those projects. Particularly, I prefer using a low order numerical scheme since I am working with compressible flows, thus with discontinuities. Working with ENO is really like a lottery. You end up adding so many uncertainties with the different method for reconstructing the fluxes that I have no idea if the real physics is captured. Of course, people leading the field do not want to discuss much about that.

How right are the observations of Julio! However, the right sense will eventually prevail. Sooner the better!

Dear professors Tapan K. Sengupta and Julio Mendez

Thanks for corporation in this topic.

I have not heared it before. Could you explain more your point of view?

I interested to high order method because of my going research fields. as far as I know, for aeroacoustic simulation we need high order methods. But you mentioned that "accuracy" is more important than "order". This sounds reasonable. Could you introduce me some papers/researches which explain this. I am interesting to know how could I increase accuracy without increasing order. this is just for spectral method or we can use in finite volume method too?

Best Regards

You are right Alireza Beiki the accuracy is more important than anything else. I want to use the words from one of my mentors "not because it shines it is gold!". Professor Tapan's book is awesome and that is a must have for anyone in the game of CFD (High Accuracy Computing Methods: Fluid Flows and Wave Phenomena). I have my own copy always at my hand. Be advised, this a very complete textbook but it not for beginners, but if you have a good understanding that will be the book. Let me finish with another quote of one of my mentors:"Not because you have a solution that is THE solution"

Thank you for your Kind words, Julio! We try our best, despite the hostile reception we have encountered all throughout. Let there be open discussions, and not hide behind the anonymity of journal-reviewer tags. I review a few and give permission to editors to reveal my name to authors. Let there be a portal, where contributing authors can post their manuscript along with the journal reviews. Let the world at least know, which editor is using "experts" in the field!

Professor Tapan K. Sengupta if you reviewed my paper I would be very happy and proud. What I said about your book is the truth. I wish I had known you back in 2008 when I started my path in CFD, things would have been very different. I believe we have to acknowledge the contribution from people like you. Here in the US things are very rough in this field, sometimes I wonder if the truth is taught or if we have to pursue the truth by ourself. Please keep up the great work and keep showing the real path.

Julio: Feeling embarrassed to read your kind words about our work. Shouldn't everyone be following the path of truth? You are young and receptive, so you can be your own guru! Let me tell you a secret: I never took any course in CFD at any level. So I did not have to carry the baggage of any "school of thought". Actually such "schools" are mutual appreciation society. Staying away from those and charting one's own path is desirable. Please pay attention to "focusing", as this is the next major issue. Specially, when exascale computing will be common place. Needless to say we have again hit a road-block. But it will see the light of day.

ansys

A wonderful conversation going on here. Professors Tapan and Mendez, God bless you both. Great to learn about your book, Prof. I will get a copy soon. Keep up the good work.

Thank you Chinedu Nwaigwe for your prayers. For my age, I am getting closer to God. In the meanwhile, feel free to post any of your questions. I would be delighted to share with you whatever I know.

Best regards,

Tapan

I think many of these high-order methods have indeed very similar discretization properties and yield comparable accuracy for comparable spatial resolutions, which has also been addressed in several publication. However, this question should be discussed in a broader context in my opinion, including also the aspect of the computational efficiency of a numerical method. This can have a huge impact (matrix-free versus matrix-based implementations, DG versus HDG. In this context, communication is actually not the point that makes the difference, but it is the serial performance of a code that may easily vary by more than one order of magnitude). Unfortunately, this aspect has not been sufficiently addressed so far. A high-order method with fast implementation that allows to compute on a finer mesh or using a higher polynomial degree with the same amount of computational costs is superior (although it is not more accurate than other high-order methods/implementations). Finally, let me mention that answering this question depends on the recent developments in computer hardware (how does the peak performance grow relative to memory bandwidth).

Niklas Fehn : Interesting comment. Perhaps it will help you to first decide what you mean by "discretizing properties"! We have given some error metrics, and hope you are aware of those. Then you will note that different higher methods are different. The clue is in knowing what you should be looking at . Hope it is not merely looking at Taylor series expansion. That will embarrass a lot of us. Happy and meaningful computing.