[ngrube]

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 N-wave 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

[harishg]

Did you check if the method worked with lower-order shock capturing schemes ? I recall a discussion on this topic in the book of Leveque for long distance propagation.

[ngrube]

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 1-dimensional 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.

[ngrube]

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 low-storage Runge-Kutta 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?

[harishg]

There is a version of the low-storage TVD R-K 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.

[ngrube]

Got it. Thanks. For anyone else who is interested, the reference is

Total Variation Diminishing Runge-Kutta Schemes
by Sigal Gottlieb and Chi-Wang Shu
Mathematics of Computation vol 67, number 221, January 1998, pp 73-85

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.

[jed]

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

@article{gottlieb2009hos,
  title={{High order strong stability preserving time discretizations}},
  author={Gottlieb, S. and Ketcheson, D.I. and Shu, C.W.},
  journal={Journal of Scientific Computing},
  volume={38},
  number={3},
  pages={251--289},
  year={2009},
  publisher={Springer}
}

a preprint is at

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 Runge-Kutta and multistep (explicit and implicit) are tabulated at
http://www.amath.washington.edu/~ketch/SSP/

[ngrube]

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 non-TVD behavior since that is much easier to fix than moving away from the WENO scheme, which I am pretty much obligated to use.

[ngrube]

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.

[harishg]

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.

[hnemati]

Hi all
I am simulating DNS of developing pipe flow with highly temperature dependent property. I saw-tooth 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