THE MULTIGRID APPROACH

      The multigrid methods were first introduced in the 70's. They are very efficient to solve general elliptic equations with non constant coefficients. Globally, with the multigrid method, you need to define your problem on a grid of pre-specified fineness and then other grids (coarser grids) are used to solve the problem (those last grids can be considered as temporary , computational adjuncts).
In fact, we can illustrate this method with a two-grid method. First of all, the residual is computed on the fine grid. Then it is defined on the coarse grid using a restriction operator or fine-to-coarse operator. At this point, the residual can be calculated exactly on the coarse grid (as there are less meshes, it is easier to solve) and then the new value is defined on the fine grid, using a prolongation operator in order to update the solution. To generalize and use the multigrid method, you just need to introduce some coarser grids and use at each step the two-grid method.

In Fluent, you can find the multigrid panel on the menu Solve. These are the different parameters you can choose:



First, we tried to run the same simulations with and without the multigrid option (with the default values of this option). The results weren't interesting as they were the same. We think that in fact Fluent always utilises the multigrid option, with the default values if you don't specify anything.
So we tried to set all the multigrid parameters to 0 in order to switch off the option. In fact, the solver refuses this kind of operation as some parameters has some minimal values and can not be set to 0.

As we couldn't test the multigrid influence, we tried to obtain results by changing some of the multigrid parameters. We just tried to change the coarsening parameters, simulating a flow with a Rayleigh number of 3692 and a precise mesh (in order to test the convergence speed-up) of 2592 nodes.

The first case had been run with the default parameters. The solution converges in 936 iterations.


The second case had been run with other values of the coarsening parameters (and default values for all the others parameters). We set the coarsening parameters as follows:

In fact, we increased the number of coarse grids which are used to calculate the solution. With these parameters, the solution converges in 102 iterations, wich is almost 10 times faster than the previous case.
 



However, we tried to run this case with  even more coarse grids but the solution doesn't converge as fast as the previous case...

As a conclusion about this part, we can say that we didn't have enough time to explore this option more precisely. It was just a first approach which needs to be continued.