Hi all,

Can anyone point me to some books or papers which give a clear description of ENO's schemes. I have several papers by Shu et al., but I am looking for something which is more engineering based.

Thanks,

J.

You may look at this paper, it has clear explanation from point of view of enginneering based, just follow the notation, then you will get the implementation, it is Weighted ENO which is much better than just ENO.

I implemented it by using C language, if you want I can give it to you.

Kim

A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics Guang-Shan Jiang, Cheng-chin Wu Journal of Computational Physics, Vol. 150, No. 2, Apr 1999, pp. 561-594

1) try to read Shu's paper: ICASE 97-65. The presentation there is quite comprehensive.

2) in another paper by Jiang and Shu on the Hamilton-Jacobia equation, there is a short but concise presentation on the WENO scheme. You may code it easily.

3) Try to contact Prof. Shu directly and ask for ENO/WENO codes.

Hi, Paul

Do you think it is possible to ask ENO/WENO code from Prof. Shu? By the way, what do you expect for the future applications of ENO/WENO schemes?

You can have a try. As I known, many researchers have ENO/WENO codes from Prof. Shu.

In my option, ENO/WENO may be the most successful shock-capturing schemes so far. But they also have their own problems: 1) too much judgements are needed to selected or assemble a suitable stencil. For most applications second-order may be enough therefore some other schemes such as the Tadmor's central schemes are usually preferred. 2) ENO/WENO scheme had been used in LES of turbulence, where it is often found that they are too dissipative. You can refer some recent papers for such usage. By the way, I really don't understand why there are so many works on turbulence simulations with low order TVD schemes. They are too dissipative!

Most papers on ENO/WENO appeared in later 80's or early 90's. You may notice that there are few papers on this topic in these years. Many applied mathematician are now enjoying Discontinuous Galerkin method, which should be much more promising for unstructured grid than ENO/WENO schemes.

Return to the topic. I believe Prof. Shu will appreciate the AUTHORIZED use of his codes. We alway want someone to cite our papers, do you think so?

Cheers !

hi paul,

as i'm new to ENO/WENO world & soon i'll start using it what shall i do. shall i contact Prof. Shu to get his code! do you think ENO/WENO applicable to single-phase only or can someone use it for two-phase flow models.

if anybody out there can e-mail me the code please do so.

student

" I really don't understand why there are so many works on turbulence simulations with low order TVD schemes. They are too dissipative! "

If you set the numerical viscosity parameter epsilon = 0 in a second order TVD scheme (Harten's algorithm), you can virtually eliminate numerical viscosity on a fine mesh. Contrary to my prior postings on the inviscid drag on cylinders, I was recently able obtain a solution at M = 0.3 that was essentially drag free (Cd -> 0). Plots of entropy increase confirmed the absence of any significant dissipation.

Using the same code, I am currently running viscous (full Navier Stokes) solutions on the 2-D flow over a cylinder for different Reynolds numbers (Re = 100, 1000, 1E4, 1E5). The underlying inviscid solver is a 2nd order TVD algorithm with epsilon set to zero. I am trying to find out exactly how much free stream turbulence is needed to introduce vortex instability, resulting in Von Karman vortex shedding. In an earlier investigation I came up with an instability criterion of Tu > 1/Re. I am trying to confirm this now through myriad tests. If I can confirm my earlier finding, I may decide to publish a paper on this.

To sum it up, 2nd order TVD schemes can be used in turbulence simulation as long as epsilon = 0. For epsilon > 0, yes, they are way too dissipative. You are better off with a first order Roe scheme based on my experience.

Altho I am quite biased, I would like to point out that in JCP 136, pg. 83-99 (1997) we presented a scheme (MP5) that :

1. beat WENO5 (Jiang and Shu's WENO scheme) in both accuracy and efficiency.

2. can be proven to be monotonicity preserving under a CFL limit, unlike either ENO or WENO schemes.

Dont get me wrong. WENO5 is a great scheme but to say it is the best shock capturing scheme is a stretch.

Ambady Suresh

Hi, Suresh,

I have read the your paper aforementioned. Using local adjustment to ensure the monotonicity preserving can be looked on as an extension of Engquist's nonlinear filter. The integration of such technique with WENO can be seen in a recent paper (JCP, 160, p452-466, 2000). With the further increasing of the formal order of the shock-capturing scheme, it seems that one has to assort other means to control the oscillations besides the upwinding mechanism.

Hi Paul:

Great that you have read our paper. I too have read the paper you mention that attempts to make WENO schemes MP following our approach.

In real life, the pursuit of very higher order accuracy cannot be discussed independently of efficiency. This is why we paid particular attention to efficiency of the scheme in our paper.

Hi, Suresh

Could you give us some engineering interest in which high-order schemes show their advantages?

YANG

Hi Paul,

Could you tell the E-mail address of Prof. Shu?

YANG

I am familiar with simple ENO methods, although I am not with WENO and MPWENO (but I have read the correponding papers in jcp). However, I would like to rise some questions about them.

1) How do you extend these MPWENO methods to multidimensional FV schemes ?? It must be noted that we are already introducing an error O(h^2) while using the mid-point rule in the integral to approxime the fluxes (in cartesian grids). To improve this (trapezoidal rule or similar) the formulation becomes quite messy (specially in 3D). It is correct to reconstruct the solution with 4-5th order WENO while we are already introducing an error in the integral of the fluxes much bigger. Of course in 1D flows, there will be no such error, and in 2D flows with aligned grids the error will be much smaller. Fractional steps as a solution?? A further step, ENO in non-orthogonal grids ??

2) MPWENO 5th + RK 4th seems a very expensive approach (CPU + memory) what do you think about ADER approach (Toro 2000) to avoid the use of Runge Kutta methods while keeping the temporal/spatial accuracy ??

3) I have not seen (yet) ENO in 3D meshes, are there any trustful result out there??

4) Is there any work in shock boundary layer interation using ENO-type methods ?? Most of the ones I have seen were performed with FD compact schemes (Lele type) with some type of corrections near shocks (ACM, filters or similar)