High order compact finite difference schemes
I'm currently working on implementing 3D DNS code for air film cooling simulation and plan to use high order compact finite difference schemes (for example those presented in the work of Lele). But first I want to test them for 2D simple stationary convectiondiffusion equation. The problem is that all examples people show in their articles are either 1D or without any detailed information on implementation. The only 2D example I found uses ADI. So the questions are: is it possible to generalize Pade type of schemes to 2D or 3D, or I have to use some kind of direction splitting, and could someone provide me good references with some details how people use high order compact finite difference schemes for 3D calculations? Thnaks in advance:)

Re: High order compact finite difference schemes
Compact schemes have been used quite a lot, in particular for aeroacoustics and combustion computations and these are all 2D or 3D. Just do a literature search. The articles are easy to find. If the grid is orthogonal the extension to a 2D or 3D formulation is straightforward, I don't see what your problem is. But, as far as I know, compact schemes have only been used for compressible flows, not for incompressible.
Tom 
Re: High order compact finite difference schemes
Thanks, Tom. I have something like several dozens of such articles on my table, but my question is still open  I need some with detailed information how to use high order compact finite difference for 2D or 3D equations, since in all articles people present in detail 1d example and nice 3d pictures, but nobody spesks about details. May be the generalization is really easy and straightforward, but for me not  that's why I ask:)

Re: High order compact finite difference schemes
Mikhail
The thing with finite difference schemes is that 2D and 3D versions are just 1D versions repeated along the other two dimensions. For finite volume schemes, 2D and 3D versions are different for truly higher order accuracy because of the challenge of evaluating derivatives multidimensionally. But finite difference schemes are simple that way. hope that explains it. ajs 
Re: High order compact finite difference schemes
Hi Mikhail,
I'm also currently working on Higher order compact schemes. Here are some references for implementing the Higher Order Compact Schemes of Lele to 2D and 3D. 1. HighOrder Accurate Methods for Unsteady Vortical Flows on Curvilinear Meshes  Visbal,M.R. Gaitonde, AIAA Paper. 2. HighOrder Schemes for Navier Stokes Equations; Algorithm, Implementation into FDL3DI and Evaluation, Air Force REsearch Lab, 1998. 3. On the Use of HigherOrder Finite Difference Schemes on Curvilinear and Deforming Meshes, Visbal and Gaitonde Hope these references are helpful. Regards, Amith. 
Re: High order compact finite difference schemes
I did my projects using high order compact finite difference for acoustic wave propagation and free convection heat transfer in cavity. I got some nice references from the Internet, here they're:
www.math.hkbu.edu.hk/~ttang/Papers/li_tang_JSC.pdf http://www.ma.ic.ac.uk/~mdixon/ http://amath.colorado.edu/faculty/fornberg/Docs/thesis1.pdf ftp://ftp.icase.edu/pub/techreports/98/9813.pdf ftp://ftp.icase.edu/pub/techreports/98/9819.pdf Pran 
Thanks a lot for the references for you both:)

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