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 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

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.

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 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.

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 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?

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.

TVD low-storage Runge-Kutta
 

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.

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

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/

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.

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
 

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