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DSS January 10, 2007 21:24

Finite Volume WENO Scheme

I have implemented the Finite Difference WENO scheme. But, with Finite Volume, I think I need to understand clearly how the reconstruction is done.

Please correct me if I am wrong.

Considering a 3D FV WENO reconstruction,

1. Obtain face average values at i+1/2 from cell averages with the standard WENO procedure (same as that of Finite Difference) 2. Obtain a reconstructed value for gaussian point along one auxillary dimension 3. For these values, again obtain a reconstructed value for gaussian points in another auxillary dimension 4. Compute Flux at these gaussian points 4. Integrate flux on the face with Gausian Quadrature rule (a simple sum in case of two point Gaussian Quadrature) 5. Use the integrated value for the flux term

So, in case of 2D, for a flux splitting scheme, with N gaussian points, we have per dimension

One reconstruction along face normal, N Reconstructions along first auxillary dimension, N Reconstructions along second auxillary dimension, One Gaussian Quadrature Integration

Thus, there are (1+N*Dim) constructions required for a Flux term. As there are left and right terms, the total reconstructions for all dimensions would be

( 2*(1+N*(Dim-1)) ) * Dim

Hope my understanding is right...

Thanks in Advance, DSS

saygin January 11, 2007 10:44

Re: Finite Volume WENO Scheme

There is a confusion in the 3rd step. Firstly, you will do reconstructions in each direction like a FD case, afterwards, you will do the Gaussian point reconstructions -in each direction- with those values you have already found for the cell boundaries (which are at the middle).

So, there will be (1+N)*Dim reconstructions, for this dimension-by-dimension structured case.

Total reconstruction steps are not changing for a flux splitting, because you are finding left- and right- state values for a cell boundary, both for FV and FD, by only one sub-reconstruction. For the flux splitting you use each splitted flux for only one state (left or right), so there the number of reconstructions are equal.

Again, I don't know any FV flux-splitting for WENO.


DSS January 11, 2007 11:13

Re: Finite Volume WENO Scheme
Sorry... it is the Riemann solver I mentioned wrongly as Flux Splitting. In this case, we need to reconstruct the left and right extrapolated values to the gaussian points. Please correct me if I am wrong.

A simple doubt. It is mentioned that WENO FV is better than WENO FD if the geometry is skewed. Does this mean that the same transformed (curvilinear) equations are treated well in FV when compared to FD, or is it the unstructured grid that was mentioned?

If I am to use a overset method, is it advisable to have a WENO FD in the background/farfield and a WENO FV near the boundary?

saygin January 15, 2007 04:47

Re: Finite Volume WENO Scheme
Yes, we need to reconstruct the left and right state values to the Gaussian points from the reconstructed values of the cell boundary centers.

They assume uniform grid spacing for the approximation of FD WENO. So, there has to be a uniform grid or a smoothly varying grid, which can be transformed by coordinate transformation for 3rd and higher orders, for FD. However, FV WENO doesn't have such a restriction in its approximation phase. It is conservative regardless of grid spacing.

With FV WENO you can use structured or unstructured decomposition, but, structured decomposition uses dimension-by-dimension strategy. Therefore, it has to be mapped to a rectangular domain for curvilinear cases (or, at least the strategy I used in the thesis requires it.)

>"If I am to use a overset method, is it advisable to have a WENO FD in the background/farfield and a WENO FV near the boundary?"

FD WENO uses fluxes and FV WENO uses the cell average values of conservative variables for the reconstruction. The reconstruction strategy differs form each other, and I haven't came across such a hybrid decomposition yet.

For an overset method, have a look at the thesis of Kurt Sebastian, but it is for FD.


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