Dear friends

Does anyone know how to use 16**2, 32**2, 64**2 grids to estimate error or oder, for lid-driven cavity flow?

Thanks in advance

Dear M,

The estimate of the error can be obtained by solving a problem on progressively refined grids. It is not necessary that you know the exact solution, the numerical solution is sufficient. What is made use of in this case is the idea of Richardson extrapolation. Consider grids G1,G2,G3 which have length scales h,h/2 and h/4. G3 is the finest grid while G1 is the coarsest. Now look at the following arguments

We can write the solution on any grid as

S_num = S_ex + C.h^p where h is the length scale asoociated with the grid and S_ex is the exact solution.

Thus, S_G1 = S_ex + C.h^p ...(1)

S_G2 = S_ex + C.(h/2)^p ..(2)

S_G3 = S_ex + C.(h/4)^p ..(3)

From these three equations (1)-(2) and (1)-(3) give

S_G1 - S_G2 = C.h^p ( 1- 1/2^p)

S_G1 - S_G3 = C.h^p ( 1 - 1/4^p)

Dividing these, we get a single equation with only one unknown, which is p, the error fall rate.

The only necessity for Richardson extrapolation is that we need solution on progressively refined grids and a characteristic grid scale, h. For structured or cartesian grids generally employed for most validation problems, like the lid-driven cavity, this is an easy task. In your case, the grids 16*16,32*32 and 64*64 are G1,G2 and G3 respectievely and if we assume the domain as [0,1]*[0,1] then

h = 1/16 on G1

= 1/32 on G2

= 1/64 on G3

Note that the length scale on G2 is half of G1 and G3 quarter of G1.

It now is a direct task to estimate the error fall rate for the problem at hand. This method can be applied to any problem provided the requirements of the Richardson extrapolation is met.

Hope this helps

Regards,

Ganesh

Is it possible to use this procedure to find out the order of converged solution by keeping the fixed number of grid points in one direction and varying in other direction by ratio of 2?

If so, how accurate it would be?

Dear Ganesh

Thank You very much. Thanks a lot. I've made "discovery" again for myself

But I still have a few questions. I found the following (Googled "RE, estimation error"):

.....(Roach 1998) with a grid refinement factor r = sqrt(3) in each of the two space dimensions p1 and p2. This factor is squared (would be cubed for three space dimensions, etc.) to determine the growth in the number of subdivisions that the 2-D domain must be discretized into for a uniform refinement of the domain. This 2-D growth rate being three, then starting with 3 subdivisions of the domain (for reasons explained below) the next refinement is to 3x3=9 subdivisions, then 3x9=27, then 3x27=81.** We pick as the refinement factor in this work for reasons explained later.

**This is analogous to a refinement factor of 2 in each space dimension, associated with a 4-times reduction in cell volume in 2-D space when halving of the point-to-point mesh spacing in each coordinate direction, e.g., in going from 2x2 cells to 4x4 cells when each cell is subdivided into 4. The growth factor is then from 2x2=4 cells to 4x4 = 16 cells, which is a growth from 4 to 16 cells or a growth factor of 4 in two dimensions or 4^(1/2)=2 in each dimension. ------------------------------------------------------- The question is:

Suppose I have velocity distribution (x,y,=0.5) for three grids.

http://gltrs.grc.nasa.gov/reports/20...000-209946.pdf

http://www.tdx.cesca.es/TESIS_UPC/AV...3Jcr03de07.pdf

----------------------------------------------------

I guess that equation with logarithm - is that what I was looking for.

Dear Ramp,

The only requirements for using RE for error estimation as I mentioned earlier is that you need to be able to `define' a suitable length scale for the grid. In case of structured grids where the points are doubled, it is isotropic adaptation where the aspect ratio gets preserved and there is no difficulty in detremining a length scale. In general we could determine it as sqrt(A). Possibly other definitions could also be tried out, based on the perimeter etc..., but for a grid with unit AR cells these would all fall back to the same definition. That is h=sqrt(Cell_area)

Now, as for your question, if we do only a refinement in one direction, which is anisotropic the AR is not preserved. So, we could in principle use a RE if we did have a length scale definition, which could again be sqrt(area). Thus while for isotropic adaptation the length scales would go as h,h/2,h/4 etc.., here it would be h,h/sqrt(2),h/2 etc ..

The same formula can be employed and I do not think there is any "loss of accuracy" in using it for such a grid. However it is true that the LENGTH scale definition is not a very unique one.

I have myself tried to compute the accuracy of a linear reconstruction procedure on grids where points are doubled in both as well as only one direction. While, I do get the expected second order accuracy in the first case, the accuracy of the reconstruction degenerates to first order as in case of any unstructured grid for the second case. Though the grid is cartesian, the result does not seem much surprising basically because of the AR effect: While AR is preserved in the first, the AR changes in the second, making it behave more as an irregular grid rather than a regular one.

Hope this helps

Regards,

Ganesh

Dear Michail,

Your question seems to contain two reports and I presume your question pertains to the equation containing the logarithm. Well, you have really done a good survey and you are right. What I had mentioned in my mail pertains to the same, only that I have applied a refinement factor r=2 and also not provided the last step, which gives the equation with the logarithm. In priciple, you could always generalise with a refinement factor say r, but rather than dividing a cell into 9(r=3), it would easy to divide it into 4(r=2), which is a more logical step. This is the reason why people go for r=2. You could also have more than three grids and then make a more better estimate. But yes, RE is restricted in use and must be employed with caution as the reports say, but for your problem there doesnot seem to be any hurdles. So you can go ahead and compute your order of convergence and make comparison with the theoretical values.

Hope this helps

Regards,

Ganesh

Dear Ganesh,

Thanks for such a detailed response. I am using structured Cartesian grid and want to know the order of accuracy. I will try in both the way.

Best regards, Ramp

Dear Ganesh

Thank You very much. You helped me a lot.

Best wishes

Michail