Hi Dear Friends,

I run my code in three meshes. From this results, I would like to know what is the order of convergence of the numerical method.

Best Regards Ferreira

ok... what is your question? how to find the order of convergence given the results on three meshes?

Which numerical method are you using? The order of convergence varies depending on which type of scheme you are using. Are you using finite volume or finite element?

~Tiger

Here is a reference book:

"Verification and Validation in computational science and engineering" by Patrick J. Roache

ISBN 0-913478-08-3

Hi Ferreira,

You can talk about convergence over several aspects: linear/nonlinear solution, relative/absolute residual, relative/absolute solution increment, convergence related with the time integration algorithm, solution scheme, among others...

In order to help with your question we have to know more about your problem... at least your convergence criteria and the order of magnitude that this criteria decays.

Regards

Renato N. Elias

Sorry my dears,

My problem is to solve the incompressible Navier-Stokes equations plus mass conservation equation on especific domain. I run my code in three meshes and compared the results with analytical solution. I noted that the numerical solution on coarse and medium meshes converges to numerical solution on fine mesh. And the fine mesh numerical solution is in good agreement with the analytical solution. My numerical code is in finite difference method; convective terms is approximated by second order upwind scheme and viscous/pressure terms by central diffrence.

Thanks, Ferreira

Dear Ferreira,

I assume that you have a siutable convergence criteria for your code and you are intersted in the steady state solution for some problem at hand. You then have solved the problem on three different grids of sizes say h1,h2 and h3.

We have,

s1 =solution on grid 1 = exact solution + C.h1^p s2= solution on grid 2 = exact solution + C.h2^p s3= solution on grid 3 = exact solution + C.h3^p

where p is the order of accuarcy.

Now by simple subtraction, we have

s1 - s2 = C*(h1^p - h2^p)

s2 - s3 = C*(h2^p - h3^p)

From these, dividing we get a relation R that has only p as the unknown and can be solved for known s1,s2,s3,h1,h2,h3.

A few comments:

1. This is basically the Richardson extrapolation procedure that is widely employed for estimation.

2. h1,h2 and h3 are characrteristic lengths of the grids, so it is necessary that your grids 'nest' in other words, we need to have regular grids such that h=h1=2*h2=4*h3. This can then be used to simplilify the relation R obtained above to a simple algebraic equation in p.

3. The exact solution is not essential for obtaining the order of convergence ie p. The solutions s1, s2 and s3 can be integrated coefficients such as Cl or Cd which are representative of the problem at hand.

However, note that the word 'convergence' encompasses a lot of things as Renato pointed out, and unless very specific it is difficult to answer to the point. I presume that the situation you are poresently faced with is something of the above kind and hence this reply.

Hope this helps.

Regards,

Ganesh

Check out the work done by Fred Stern (Iowa) in this area.

If you have the exact solution then compute the error in some norm (like L1, L2, Linf) and plot error versus log(h) where h is a measure of the grid size. Then the slope of the resulting line will give you the order of accuracy. You can do a least squares fit for error versus log(h).