Dear friends,

I applied unstructured multigrid method to Laplace equation, I found the solution converged effectively. But when I applied the sam method to Euler equations, I found it had no use. Sometimes it caused the solution divergence. Could you give me some suggestion on unstructured multigrid method? Thanks in advance.

Jian Xia

Hi Jian,

The answer I posted went away with the disk crash yesterday, so I am repeating myself here again.

From my own experience, multigrid methods work fine for elliptic equations (with a diffusive term - second order derivative in space) but does not work for hyperbolic equations (with only advective terms - first order derivative in space). I have never seen examples of hyperbolic equations solved with the multigrid. Usually one wants to solve implicitly elliptic terms (diffusive), since the restriction on the time step (for time dependent problems) is extremely strong (stronger than the CFD restriction due to advective terms). The implicit solver requires a matrix inversion, which is extremely time consuming and to avoid that multigrid methods are used.

I never worked with the unstructured multigrid method, and my knowledge of multigrid is limited. So it would be good if someone could comment on your question too.

Cheers,

Patrick

(1). I think, the elliptic equation tends to produce soft solutions. (2). On the other hand, the Euler equation tends to produce hard solutions. It resists the change in time and space. (3). The soft solution depends on the boundary conditions all around, a corase mesh can transfer the information faster than the fine mesh. (4). The hard solution depends on only certain portion of the boundary. It is directional. The use of a coarse mesh to bring in information from non-related boundary, can not produce healthy solution.

Hi, There is a strong work done in the projet SINUS at INRIA on multigrid method for compressible inviscid and viscous flows. You can see the the web site

http://www.inria.fr/RRRT/publications-eng.html

and search multigrid.

Good luck.

Farid++

hmmm, seems strange. multigrid was developed for elliptic problems but i know that Antony Jameson (and many others) have used multigrid with much sucess. a search of jameson's papers (say in aiaa or j comp physics) might help. the techniques he used is called Fast Approximate Storage (FAS). actually i got a ref to one of his papers. it is: Solution of the Euler Equtions by a Multigrid Method in Appl. Math. Comput. vol 13 pp 327-356. also the AIAA journal of aircraft had an issue last year (i think it was december) on Multidisciplinary Optimisation in which Jameson has 3 papers and i think he used a multigrid euler scheme in two of them so some good references can be found there as well. hope this helps

Sorry, I have not been following this forum for a while, that is the reason for this late reply. At our institute we use unstructured (agglomeration) multi grid for the 2D Euler equations. Our multi grid has been developed during a close cooperation with 'Projet SINUS' at INRIA Sophia Antipolis. The code is publicly available from http://www.vug.uni-duisburg.de/MOUSE

While the multigrid method was initially developed for elliptic problems, it can and does perform exceedingly well for hyperbolic systems in particular the Euler equations, and from my perspective has become a routine element of any explicit time marching code for the Euler equations.

Jameson's work through the 80's documents its development and refinement quite well.

Bob

Hi Bob,

Thanks for the update.

You mentioned that MM has become a routine element of any EXPLICIT time marching code for the EUler equations.

I always saw the MM actually as an alternative method for the simple matrix inversion in IMPLICIT time schemes (at least for the elliptic terms).

Could you shed some light on this point? Thanks.

Patrick

Depending on your perspective, implicit and explicit solves are not terribly different. For example an explicit Runge-Kutta integration step can be seen as being very similar to a Jacobi iteration in a typical matrix solve coming from an implicit discretization. From this perspective I'm quite sure the analogy goes through for multigrid schemes in either case. In the implicit case it is seen as a matrix solve acceleration method and in the explicit case as a convergence to steady state acceleration method. The essential foundation of error estimates and mode damping works either way. For hyperbolic equations you also get the additional benefit of accelerated mode propagation on coarser meshes.

Bob