WENO for traveling/steepening acoustic wave
Hello,
I am puzzled by the failure of a well debugged and validated WENO finite difference code in the case of a traveling acoustic wave propagating over a long distance. I expected the originally sinusoidal wave to steepen into a sawtooth or Nwave pattern. What I see instead is the development of shocks (where they should be) plus somewhat weaker but still significant shocks approximately halfway in between, so I get twice as many shocks as I should. I might faintly recall seeing a discussion in a journal article about the inability of WENO to handle this case. Or maybe I just dreamed that. I have been searching with all the keywords I can think of and can find nothing. Any help would be appreciated. Thanks, Nathan 
Did you check if the method worked with lowerorder shock capturing schemes ? I recall a discussion on this topic in the book of Leveque for long distance propagation.

Leveque
Thanks for your reply.
No, I have not tried lower order shock capturing on it. I need high order since this code will be used for turbulence. This case is just a test case I wanted to use to see how a new turbulence model behaves in the presence of these shocks. (But this test case is 1dimensional and hence laminar of course.) I will look for something relevant from Leveque. I have requested his finite difference book from another library. Will post again if I learn anything. 
Time integration
It just occurred to me that if I end up with twice as many shocks as I should have, then the total variation is not diminishing. I have been using a Williamson 3rd order lowstorage RungeKutta scheme for time advancement, and now that I think about it, I am not certain that that is TVD. I know many people use RK with WENO methods, but maybe the low storage RK that I am using is not suitable?

There is a version of the lowstorage TVD RK scheme which was implemented by Shu and Gottelieb. If you google for it, you might find the report. The book of leveque is " FVM for hyperbolic problems". He talks about the possible failure of linearized Riemann solvers such as Roe one in that. I am not exactly sure though if thats the exact thing that you might be looking for.

TVD lowstorage RungeKutta
Got it. Thanks. For anyone else who is interested, the reference is
Total Variation Diminishing RungeKutta Schemes by Sigal Gottlieb and ChiWang Shu Mathematics of Computation vol 67, number 221, January 1998, pp 7385 Also, the Leveque info is in section 15.3.7, "Failure of Linearized Solvers." Not sure yet whether that is pertinent. I plan to try a TVD RK scheme first. Will post with the results when done. 
Better reference
This field (now known as "strong stability preserving" time discretization, WENO is not strictly TVD by the way) has evolved significantly in the last decade. A good current reference is
Code:
@article{gottlieb2009hos, http://www.amath.washington.edu/~ketch/David_Ketcheson/Publications_files/SSPreviewFINAL.PDF if you don't have access to the journal. A selection of optimal SSP RungeKutta and multistep (explicit and implicit) are tabulated at http://www.amath.washington.edu/~ketch/SSP/ 
Thanks, Jed, for those very helpful references and for cluing me in to the term "SSP." As you say, it is true that WENO isn't TVD of course. I was just hoping that the time integration was responsible for the nonTVD behavior since that is much easier to fix than moving away from the WENO scheme, which I am pretty much obligated to use.

Problem seems to be with WENO, not RK
Well, I've coded a TVD RK scheme and get almost identical results. I would say that my problem is with trying to use WENO for this.

Start with first order upwind, Godunov schemes and then try higher order scheme. I think its also possible to reduce order of ENO scheme by using a smaller stencil size.

toothy turbulent structures
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Hi all
I am simulating DNS of developing pipe flow with highly temperature dependent property. I sawtooth this structures in my enthalpy and velocity fluctuations. My simulations is highly resolved and I know that they are physical. I could not find litretures to get help from it in this issue. Can somebody help me in this case? or introduce some refs.? Good day guys 
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