ENO scheme: (Shu and Osher)
 

Hi all,

I am implementing an ENO's scheme in a F.V. method. I am trying to write a simple 1-D code at present. However, I know that I need to use the divided differences (I want to use a 3rd order accurate polynomial) to calculate the "smoothness" and hence the relevant node to add to my stencil. However, lets say my fluid is going from left to right, and I have 100 cells. Now, if I want to implement a scheme which uses the 3rd divided differences, then, say, if my current cell is cell 98 (or 99), I don't have enough cells to the right of cell 98 (or 99) to complete the calculation of a 3rd divided difference. My question is, if this is the case, then do I have to use an ENO's scheme of lower order at the right boundary, and also, how do I apply the ooutflow B.C. The former question is the most pressing. Thank you all. John.

Re: ENO scheme: (Shu and Osher)
 

Use more than one ghost cell outside the domain (say, 101, 102, 103), and fill them according to your inflow / outflow B.C. You may have to be creative there, i.e extrapolate gradients, use Riemann invariants, etc.

Re: ENO scheme: (Shu and Osher)
 

Hi Alex,

Thanks for you help. I was afraid I might have to use ghost cells. I am afraid I don't know alot about riemann invariants apart than that they are the characteristic variables for the euler equations. I am using a simple linear consevation law.

Can you give me more information regarding you comments? If I create ghost cells, I will have to give them values, and this will influence what goes on within the domain. Have you any references which may be of use to me? Also, Would it be possible to use polynomials of lower order as I approach the wall, such that 2 cells from the wall, I have a polynomial which is linear etc...? I can use a polynomial which is entirely upwind at the last cellsnear the outflow....but when it comes to calculating the divided differences up to say the 3rd divided difference, I need cells outside the domain for the calculating the divided differences starting at cells n, n-1 and n-2, where n is the total number of cells. Could I simply use lower divided differences close to the exit to choose which cell to add to my stencil, or does this dramactically affect the accuracy? Any references or ideas?

Re: ENO scheme: (Shu and Osher)
 

J.K.

Although I have read a few papers in the past on ENO, I am not that familiar any more with its polynomial fitting. I usually just stick to 2nd order TVD schemes, which become first order near extrema. Personally I would not bother with Riemann invariants. Other, simpler methods can be just as effective.

For a 1-D problem (pipe or shock tube) you only have two types of B.C.'s, wall or open end. For the open end, I would fill the ghost cells with ambient conditions. For the wall I would use a reflection B.C. To fill more than one wall ghost cell you can extrapolate gradients from the interior. An example of gradient extrapolation can be found on page 62 of my dissertation, which can be downloaded from my website,

www.microcfd.com/papers.htm

My dissertation also includes a big chapter on the shock tube problem, if that is what you are working on. With TVD and ENO schemes, there are a number of intermediate quantities which also have to be extrapolated somehow. These quantities are denoted in 'Fraktur' type set in my dissertation. With those intermediate quantities, one has to be 'creative' at times near B.C.'s.

I hope this helps!

Axel