Hello everybody:

I would like to point out some questions about algebraic multigrid (AMG) applied to an algorithm that solves the uncoupled Navier-Stokes equations with pressure poisson equation with a finite volume method (FVM).

(I have been working with the lid driven cavity benchmark)

I only apply AMG over the pressure poisson equation and, in each iteration, I must calculate the Galerkin operators since the coefficients of the pressure matrix changes each time, so it takes a long time. And although, the number of iterations for the convergence goes down with AMG, the total time of the algorithm increases.

So I think, AMG is better for FEM (because the Galerkin operators only have to be calculated one time) instead of FVM, am I right, if not why not?

Thank-you very much for your time and for your replies. I would like to say that the information about multigrid in this page is really good,congratulations! Isa

Why do the coefficients change? Between grids or between iterations?

From my experience, multigrid can work almost perfectly for the driven cavity test problem. I implemented a Full Approximation Scheme (i.e. solving for u, v, p together on all grid levels) in an industrial code in the late 80s. If I recall correctly, convergence required about 6 or 7 outer iterations to enter single precision roundoff and was independent of the size of the grid. I tried to find the report to check Reynolds numbers but do not seem to have it anymore. My guess is the results were for lowish Reynolds numbers but the scheme upwinded and so it might have held for higher Reynolds as well.

I should add that for real industrial problems with turbulence models the multigrid scheme was more efficient than the ADI-like scheme it replaced but the convergence rates were no where near as good as those for the laminar driven cavity tests.

I think the report is gone but I have found my notes. If you want some numbers to check against drop me an email.

Hello Andy,

The coefficients change in each iteration, I mean, I apply SIMPLE with FVM (finite volume method) and when I calculate the intermediate velocities (u*,v*), I calculate the pressure correction matrix and this one depends on the velocities coefficients so, that is why it changes each time I calculate the velocity. And to calculate the Galerkin operator I have to work with the new coefficients' matrix, don't I?

If you don't mind I would like to have your notes or your results to check if I am doing the right thing. Did you use FAS with FVM and with Fortran?

Thank-you very much for your time and your reply. Isa

The coefficients for the pressure equation (divergence of momentum equation) are purely geometrical. The dependence on velocity comes from the approximation SIMPLE makes for the relationship between pressure correction and velocity correction. Other pressure correction schemes make other approximations and many do not involve velocities. The simplest is to simply retain the time derivative in the momentum equation to express this approximate relationship. It works as well and sometimes better than SIMPLE so long as you get the size of the time step right.

Having said that, the evaluation of the pressure correction coefficients in a SIMPLE scheme is usually trivial because one simply reuses the diagonal coefficients from the momentum equations. There is no point reevaluating these coefficient, as you seem to be doing, because the relationship is approximate.

If you are looking to keep the scheme longer term it may be wise to look at extended pressure correction schemes like SIMPLER and relatives which require more computation per iteration but solve the equations to a better degree and are usually more robust.

The implementation was mainly in Fortran but with a PL/I frontend when used by the company. I cannot recall if it was a finite difference of finite volume implementation which makes a difference to the scaling of the residuals and coefficients between the grids. Mass conservation between the grids was also important but, I think, you should get that by manipulating the coefficients directly.