Richardson extrapolation
Does anyone have experience with applying Richardson extrapolation to compute estimated errors for the whole flowfield? I'm not referring to globally integrated quantities such as lift/drag, put pointwise application of Richardson extrapolation at each (x,y,z) on the finest grid.
One must have solution data at the same (x,y,z) from each of the meshes used, so some interpolation is required. However I haven't seen this detail described well in various papers. Many don't mention it at all. Any comments? Thanks. 
Re: Richardson extrapolation
Dear pc,
I have not used RE for error estimation as such, but as far as your question is conerned, this is one way you can do it. Let A0 be the initial grid. Let A1,A2,... be the finer grids. Thus any cell in A1 would have a "parent" in A0, any cell in A2 will have a parent in A1, and so on. Thus, for any cell on the finer grid, one possibility is to assign the values of its parent on the coarser grid. If you are intersted in transferring the information from finer grid to a coarser one, you just need to assign to the parent cell on the coarser grid, an average of the values of its offsprings on the finer grid. Since there is an inherent nesting of cells when RE is being used, this concept should work reasonably well. This kind of idea is used in adaptive strategies to map solution from a coarser grid, to an adapted finer grid, to obtain a good initial guess to solve on the adapted grid and enhance convergence. Hope this helps Regards, Ganesh 
Re: Richardson extrapolation
Ganesh,
I have used RE for global quantities and now I'm looking to see how I apply it to examine local errors. Your suggestion is consistent with a paper I just read over the weekend, in which the finer grid solutions (A1,A2) are transferred to the coarse grid (A0), and the extrapolation is visualized there. However I had thought the postprocessing would be the other way around, that is, coarse/medium grid solutions would be transferred to the fine grid. My thought being that the fine grid should be closer to the grid independent solution, and therefore an extrapolated error on the fine grid would be more meaningful than that on A0. However when I attempted this I got a lot of noise in my plots of error. Any other comments? pc 
Re: Richardson extrapolation
Dear pc,
The finegrid solution, which when proven to be grid independent serves as a reference for comaprison. In case of operations involving multi grid, the transfer of data from the finer grids to the coarser grids and back is common enough( A nice article can be found on http://www.cfdreview.com/article.pl?.../11/28/2217256). In fact, when the data is transferred to a coarse grid, we are in fact converting low frequency errors on the finer grid into high frequency errors on the coarser grid, on which schemes like the GaussSiedel work very well. When transferring back the data, as the article points out "...The interesting thing to note is that as we solve the algebraic equations on the coarse grid and then impose the new values back on the fine grid we introduce high frequency error into the fine grid...". This is possibly creating the noise in the error plots. However, the noise is also dependent on the level of coarseness. In my own research, although I do not explicitly use RE for error estimation, I do transfer data from a coarser mesh on to a finer mesh. The important thing to note is that the finer grid and the coarser grids do not differ by more than one level( that is to say that the finer grid is an immediate nesting of the caorser grid under consideration). In fact, transferring data from a coarser grid on to a finer grid as also the reverse should be fine enough, the choice being made depending on the purpose you are looking at, the strategy( how you actually implement it) and also the coarseness that comes into picture. In fact if the mapping was from an A0 grid to an A4 grid directly in a single step, it is not surprising to see high frequency errors in the domain. A better option could be to do this through intermediate steps. I sincerely hope that I have not deviated from the question you had posed and have not confused you to any extent. Hope this helps Regards, Ganesh 
Re: Richardson extrapolation
Ganesh,
I've worked with multigrid before, and your arguments are sensible in that context, and in light of what I'm seeing. I guess the basic difficulty I'm having is in understanding how some of the papers I've found are able to report local applications of Richardson Extrapolation and/or a Grid Convergence Index (GCI) along an entire J line in a structured grid, for example. Or similarly, a 2D contour plot of GCI in another example. pc 
Re: Richardson extrapolation
Dear pc,
I suppose the problem is the way people report issues. I am not really sure as to what can be comprehended by repoting the GCI along a given J line. Possibly, it is one way of saying that the errors in certain portions of the grid have attained acceptable levels or otherwise. The same could be the case with GI contour plot. It could be one's way of representing the results and trying to explain thier observations. This could possibly also have been achieved by just plotting the error levels itself, but the authors could have thought otherwise. In any case, I wish you best in your endeavours. Regards, Ganesh 
Re: Richardson extrapolation
Thanks for taking a crack at it!

anyone else?
Would anyone else have thoughts on this topic?

Re: anyone else?
to apply richardson you will have to define an additional variable such as
V(n) = (U(n+1)  kU(n))/(1k) 0<k<1 where U(n) is the variable (velocity component) at iteration n. You will have to extrapolate U(n+1), which is what the cfd code does already. k could be your underrelaxation factor. What you can do is check your dU(n) = (1k)V(n) at each iteration. the maximum convergence efficiency is when dU(n) or V(n) is monotonic decreasing in absloute value. Is this what you want to do? 
Re: anyone else?
No, that's not what I mean. This is done as a postprocessing step, not within the iteration as you suggest.
Say one has 3 converged solutions on 3 different meshes from a grid refinement study. The intent is to extrapolate to an "exact" result in the limit as grid spacing goes to zero, thereby giving you an estimate of solution error at each grid point. It provides an assessment of gridindependence. What I'm looking to do is apply this concept to each local point, not simply on a global quantity. 
Re: anyone else?
it is not richardson method then?

richardson extrapolation example
hello!
I am trying to get a value from an independence mesh test, and I was told to use richardson extrapolation. If I only have a table with mesh values and the force value. how do I do the extrapolation? I am not able to find suitable examples anywhere.. those are an example of the data. By eyesight I can guess that the value of F on N to infinite will be close to 1.421.415. But how I get the value analitically?? N 10 20 30 40 F 1.5984431 1.4780298 1.4446029 1.4293989 Thank you in advance Puigar 
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