Can anyone suggest references on higher order Finite Volume method on distorted meshes?

Thank you in advance,

Sergey Smirnov

Hi,

As far as I know, this is not a very popular subject because a) the principle doens't fit very well with the traditional FVM and b) in fluid flows (where FVM is strongest) you don't get much benefit because the solution is not smooth.

Here's a reference:

@Article{Peric:FOURTH, author = {Lilek, \v{Z}. and Peri\'{c}, M.}, title = {A fourth-order Finite Volume method with colocated

variable arrangement}, journal = {Computers and Fluids}, year = 1995, volume = 24, number = 3, pages = {239-252} }

Dr. Jasak, this is true in the traditional CFD area but in computational aeroacoustics where high order methods are required FVM is the coming thing. you'll find that most people in the field use finite differences but i'm sure that FVM will catch on for some of the same reasons that it has in traditional CFD. i'm also convinced that it will also become very popular in the closely related LES field.

Hi,

Thanks for the response. I suppose you guessed by now that I'm from a commercial CFD world, where (as we obviously agree) my statement holds. I've been looking at computational aeroacoustics recently and what makes it so awkward for me is the fact that, as you say, one simply cannot do a good job with second-order methods.

The thing is that for a higher-order FVM method you have to choose between having more than one integration point in each CV, which makes face interpolation a bit awkward and (I think, based on my FEM experience) does nasty things to your matrices: no more diagonal dominance control = no more iterative solvers !!

The other option is to "extend" the molecule, like for example in QUICK convection differencing scheme, which is OK for structured meshes, but in the world of mumbo-jumbo meshing I have to deal with it's a bit more difficult. The crucial point is that even this approach will screw up your matrix structure. Form where I'm sitting, my major advantage over FEM is precisely the fact that I can effectively tailor my matrix to what I want it to be and I'm not sure this is possible in higher-order FVM. If someone out there would care to prove me wrong, I'd be more than happy: my Finite Volume Stress Analysis work is begging for a nice, compact stable and bounded higher-order scheme.

Now, LES: in the romantic years (early 90s, as far as Finite Volume LES solvers are concerned), the Stanford codes were something like 10th order in time and 6th order in space (we are actually talking finite differences here, but that's the best example I have). People swore to me (besed on their DNS work, I suppose) that you can't get a decent LES result without that sort of accuracy. But!! LES is a very "spiky" calculation and you're always fighting to resolve the smallest possible flow features (a vortex on a 2x2 mesh!). If you use higher-order numerics you throw a good part of your resolution away, and we are talking of multi-million cell jobs here! For this reason (to my knowledge) most of the LES community is back on second-order again - it just so much cheaper.

As for DNS, that's a different story... but let's do it some other time.

Hrv

i can't really comment on your matrix discussion. most CAA codes use explicit time stepping usually R-K so matrix structure isn't an issue. i wouldn't feel comfortable using a second order accurate LES code. CAA and LES are right now on a converging path. i think four years from now most cutting edge work is going to be direct noise simulation from LES predictions with Kirchhoff or Ffowkes-Williams Hawkings equations used to propagate to the far field. in CAA the DRP 4th order schemes (derived by Tam) which use a seven point stencil are most popular because they give better wave propagation characteristics than sixth order accurate seven point stencil schemes. obviously these differencing schemes are also good for LES. Most LES i've seen uses 4th to 6th order accurate differencing with a wide variety of upwinding (ENO, TVD) or artificial dissipative central differencing schemes. along with various R-K schemes these seem most popular. R-K schemes are almost de rigeur because of the need for parallel computing - it is difficult to implement implicit schemes in parallel. also most implicit schemes are first order accurate or approximately factored. however a PhD student here (Penn State) named Chingwei Sheih (his thesis shouls be out soon) has used dual time stepping to produce an implicit scheme (multigrid R-K is used for the steady state inner time loops) which is easy to parallelise. he uses a hybrid RANS/LES approach called DES (D-discrete) that uses a spallart-allmaras turb model (you could use anything else eg k-e/k-omega) with or without wall functions to simulate flows basically at the LES level and at the same time producing the pressure fluctuations to produce direct noise simulation. very impressive stuff. it virtually makes all the other CAA stuff we've done here redundant. the disadvantage is that he still uses a structured mesh. i have seen unstructured mesh LES work from a guy at Hitachi i think it was and you can also look in the INRIA reports database (www.inria.fr) for some work on this. i think CAA is going to be the main impetus for continued LES development because it combines the noise generation and near field propagation in one but at least second order time accuracy and fourth order spatial accuracy is necessary.

"If you use higher-order numerics you throw a good part of your resolution away"

I don't understand what is meant by this. Resolution in what sense?

Bob

Me again!

Take a look at

http://monet.me.ic.ac.uk/publications/les.html

and you'll find the work I'm talking about. The code in question (have a dig, you can find out a lot about it) is arbitrarily unstructured second order and implicit and massively parallel (I'm talking up to 1024 processors!) with message-passing. Also, LES with 6-th order is all very nice but I don't have a structured mesh!!!!! The guy who ran lots of this (second-order unstructured LES) stuff is called Christer Fureby - if you search on-line publications databases, you'll find a lot of his publications.

I appreciate your problem with second-order numerics _ I myself recently had an example of a "second-order" code that would produce rubbish in LES setup. Please take a look at the results: they are really good (and note that I don't have my name on the paper, so no axe to grind). I think this work may really interest you, at least as a viable alternative to the valid approach you're currently using.

Also, I would just like to make a quick point: parallelisation of ICCG and even multigrid solvers is a solved (and published) problem and lots of people have done it. I appreciate that there may be advantages to be gained from Runge-Kutta because you're a-priori limited on the Courant number for accuracy reasons, but a decent setup of an implicit code (in my opinion) is comparable.

Hrv

Hi,

I hope you'll forgive me to talk in the FEM, language - I just find it more convinient to explain and the principle is general.

Imagine I've got a feature of nominal size 1 and I'm trying to resolve it in two ways: a) by increasing a local order of a finite element (linear, quadratic, cubic, etc) and b) by splitting the element in half in each direction.

Let us first consider a "smooth" feature, say a power-law function. As I increase the order of interpolation, I pick up more and more of higher derivatives and me error reduces. Once I get enough derivatives into the element, I get the exact answer. However, if I keep subdividing the element and keeping the "shape function" linear, I'm just approximating smaller and smaller bits of the function with a straight line, right? So, however many elements I put in I'll never get it quite right, although the error does indeed go to zero. This behaviour is typical for a function where higher derivatives go to zero (smooth).

Now, let us consider a step-function. When I start doing a), I actually mess up the solution, because my polinomial representation wants to oscillate around the discontinuity. In a nutshell, The Taylor series is not the nicest way of dscribing a step. When I start doing b), I still don't get the right answer, but the whole thing is much better behaved.

In conclusion, I'm better off doing low-order approximation with lots of elements of "spiky" fields and for smooth fields I'm better of doing higher order (there's things like super-linear convergence when you combine the two etc. but this is the principle).

Now, let's consider a single vortex in LES: if I've got a low-order numerics I'm more likely to get for velocities going round in circles (however silly that might look when the vortex is small enough); if I've got a single higher-order element, the poor vortex gets smeared because I'm trying to "describe" it in terms of higer derivatives in Taylor series (see above).

The short answer to your question is: for the same money (number of unknowns) I can resolve more sall vortices with a second-order scheme than with a higher-order scheme (because the field is spiky).

If there's still a problem, please reply and I'll try to explain again.

Hrv

Michel Delanaye wrote a very nice PhD thesis on quadratic polynomial reconstruction schemes on arbitrary (eventually distorted) meshes. The thesis can be retrieved from his web page: http://george.arc.nasa.gov/~delanaye/home.html

Ok, I figured you must have meant on a "per cost" basis. It seemed to me you were implying the comparison for a fixed mesh.

As for your conclusion about 2nd order being a good trade-off point for non-smooth fields, I agree with you. But I agree based on experience with computing shocks, where the non-smoothness is even more pronounced. It is interesting to me that the same thing happens while resolving vortices.

Bob

Hi,

another very important factor is that FV requires both interpolation of fluxes at the integration points AND approximation of the surface and volume integrals for each volume. The order of the entire scheme is affected by both approximations. Surface integrations of order higher than 2 result in increasingly complex stencils. The use of a 3rd order approximation for fluxes with a second order quadrature results in a scheme which is overall 2nd order. This is quite well described in Ferziger and Peric's book and the work by Peric et al. described above.

Regards

Duane