A question: Is there any disadvantages with compact difference scheme? If the answer is `no', why is compact scheme not so popular?

Compact schemes are known to provide spectral-like spatial accuracies. Spectral schemes (FFT mostly) are often used for uniform grids. For non-uniform grids, you can either use Chebyshev-spectral or compact schemes.

Much like spectral methods, compact schemes are highly non-local. If there are narrow regions with strong gradients in the flow (like thin exothermic flames, multi-fluid interfaces) it is not a good idea to use these methods. They tend to create dispersive behavior.

However, most compact schemes involve few nodes thus are high local, that is indiated by it's name. In fact, what I want to know it that since compact schemes are high accuracy in nature, why there are so few applications of them? Many guyes in this forum said `i use upwind scheme for advective terms and second-order center difference scheme for viscous terms ...'

Compact schemes are very expensive as the derivatives are also taken as unknowns. Eventhough, there has been a publication of using this derivative for the viscous terms. It necessarily involves inverting matrices to calculate the fluxes. I would also extrapolate that it would become harder with arbitrary grids (as is true with other schemes as well).

I agree with Paul that compact doesn't necessarily mean central or upwind. A compact scheme can be either!

chidu...

For a given field F, you compute the derivatives at all points along a line (in a vector F_x) using a linear system A F_x = b. Both the matrix elements in "A" and vector elements in "b" are computed using compact stencils but that does not mean the derivative computation has a local character.

F_x = inv{A}.b

inv{A} most certainly does not have a compact support.

As to why compact schemes are not used often, one likely reason may be that they are not as accurate on curvilinear grids as they are on cartesian meshes. Also, building in the BCs into the linear system (when using a implicit solver) is not too convenient although not impossible. They also have a central-like nature which leads to dispersive errors whereas the dissipative odd-order upwind/upwind-biased are often considered more stable for advection.

Kalyan's explanation is very illustrative. In that sense, many implicit compact schemes may not `compact'. However, these do exist some explicit compact schemes which don't necessary lead to a linear system. This maybe another question.

Thank you, kalyan and chidu.

Can you point me to references for 'explicit' compact schemes? My readings include papers by Lele and Krishnan. I was under the impression that the compactness of the stencil for a given truncation error is acheived through the implicitness of the derivatives!!

chidu...

Hi Kalyan,

I agree with you about the non-locality implicit in the inversion process. Also, like I said it would be cumbersome on arbitrary grids. But I think, although most of the use of compact schemes have tended to be central-like, it is possible to derive compact schemes that upwind-based, no?

chidu...

I am sure you can build in some upwind type nature into the compact schemes.

I am no expert in schemes let alone the compact schemes but when I do LES I do have to care a great deal about the numerical errors. From what I see, compact schemes have been very attractive for temporal simulations but for spatial problems, the outflow BC is hard to handle. This is yet another problem with these schemes. As Chidu suggested, if some sort of upwinding were to be built-in, then perhaps this problem can be handled more reasonably.

The man you want to investigate is Andrei Tolstykh. He is a Russian academist who has investigated non-centered compact schemes for fluid dynamins for much of his life. He has a book, translated to english, entitled "High accuracy non-centered compact difference schemes for fluid dynamics applications" which describe how to get 3rd and 5th orders of accuracy from 3 point noncentered stencils. Hope this helps.

One can't be expected to buy a book just for non-centered compact schemes (especially if he/she does not work full time with scheme development). Can you provide us with us references in the form of papers.

Thank You.

Most of his papers are in Russian. These papers can be found in Russian journals such as Zh. Vychisl. Mat. Mat. Fiz., or other equally clear titles. However, I would recommend searching author databases for "A. I. Tolstykh". In scanning the index of the book, it appears that the english translations of some of his papers can be found in "USSR Comput. Math. Math. Phys." Try looking in the following Vol, year, page listings:

18, 1978, 134-147 16, 1976, 92-100 21, 1981, 339-354 25, 1985, 127-134 29, 1989, 183-194 amongst others. Once again, the easiest and most cohesive method is to check out his book as I listed earlier. The book seems straight forward to this self-taught CFD student, and will provide the most useful instruction, as opposed to hunting through journals.

I think compact scheme is very accurate for hyperbolic equation (see Ghrist's dissertation, The University of Colorado). Obtaining the derivatives is simple, you just solve tridiagonal matrices by using Thomas algorithm (for 4th & 6th order).It is difficult to handle the BC for the derivatives.

It's true that compact scheme can be central or upwind. (the former is popularised by Lele in a 1992 JCP paper, the later is dicussed briefly by Hirsch in "Numerical Computation of Internal and External Flows vol1, which should be found easily in univeristy libraries).

Looking at Lele description, one can see quite clearly that the uniformity (as well as regularity) of Cartesian grid plays important role in supressing/removing higher order error.Essentially in compact scheme the multiplier of neighbouring derivative is set in a way to minimise error of the original/base explicit scheme. Thus one get higher order of accuarcy for a given compact stencil.

It means that its application for curvilinear or arbitrary structured grid is not easily done. (see Smirnov, Lacor and Baelmans in ECCOMAS 2000 for curvilinear case).

I did some study in solving linear problem of convectiong a double sine wave profile (2D case) in unstructured grid with the following conclusions: 1. in term of cost the scheme is as expensive as implicit solver. (generally less than 10 SGS iteration is enough to obatin the derivative in one time level) 2. accurate result can be obtained if the grid is formed by triangulating uniform structured grid as the dispersive error is minimised. 3. the error revert back to original/base explicit scheme if the grid becomes completely irregular such as the one produced by Delaunay triangulation since it is difficult to chose multiplier that minimise error in all direction (notably the cross derivative term)

In the light of the result it is possible to see why that compact scheme may not be so popular. The coding complexity is not justified for arbitrary grid. Having said that, it is not a final word by any means. Further research on this topic is still being carried out

Cheers