From: jzhang@cs.engr.uky.edu (Jun Zhang) Subject: Re: help with MULTIGRID Date: 10 Mar 1999 15:07:14 GMT Newsgroups: sci.math.num-analysis Keywords: Using multigrid solvers for hyperbolic CFD problems Multigrid solvers are problems dependent and have limited flexibility in terms of applications. You essentially have to write different codes for different applications (at least modify some components of a existing solver). If your solver is for elliptic problems, it will break down when the CFD problems become "hyperbolic". You just need to modify your code (change relaxation scheme, intergrid transfer operators, etc.). On the other hand, Krylov subspace methods are for general sparse matrices. They can usually tolerant variation of matrices, although the convergence rates may be affected. Here are the general ideas about the two classes of methods. Multigrid method is optimal for the particular application. If it works well, it will beat the Krylov subspace methods such as BiCGSTAB. However, its applicability is a problem and it is time consuming to modify a multigrid method for your particular applications. Krylov subspace methods can usually be used to solve much more problems, particularly with suitable preconditioners. However, for particular problems where a true multigrid method can be designed, preconditioned Krylov subspace methods cannot beat multigrid method. So if you are writing a code to solve flow problems varying from elliptic to hyperbolic, I would like to suggest to use a preconditioned Krylov subspace method. Unless the performance is critical and you have time and patience to develop a true multigrid method for your applications. Best regards, Jun Zhang ---------------------- In article <36E65AF1.C1BD98B0@paris.sgi.com>, Jean-Pierre Panziera wrote: >was told that Multigrid solver would just break down when multi-phase >CFD problem becomes "hyperbolic" >while regular iterative biCGStab would would converge ... > >Why would this be ? >Any reference ? >Any advise ? > >many thanks > > Jean-Pierre Panziera - SGI France - jpp@paris.sgi.com ********************************************************************** * Jun Zhang * E-mail: jzhang@cs.uky.edu * * Department of Computer Science * URL:http://www.cs.uky.edu/~jzhang * * University of Kentucky * Tel:(606)257-3892 * ============================================================================== From: fwang@math.uci.edu () Subject: Re: help with MULTIGRID Date: 12 Mar 1999 02:16:15 GMT Newsgroups: sci.math.num-analysis Generally speaking, if you order the nodes in the downwind direction, it improves the multigrid performance in the nearly hyperbolic case. If you have the time and energy to write your own code, a special block Gauss-Seidel method might just do the job. Basically you group the nodes in the cross-wind direction into one block, and order the blocks in the downwind direction. The convergence can be very fast. Feng Wang In article <36E6C379.2A217ABA@mrco2.carleton.ca>, Ed Hughes wrote: >I agree with Jun Zhang, but my experience has been that Krylov solvers >(I've used GMRES) become much slower to converge as the problem becomes >more nearly hyperbolic, i.e. as the skew part of the matrix becomes >dominant. I don't know of any preconditioner that works really well for >this case. If anyone else does know of a preconditioner for GMRES that >handles well the near-skew case, I would be very glad to hear of it. >--Ed Hughes >Edward Hughes Consulting >Ottawa, Ontario > -- ~{KDTB~} http://www.math.uci.edu/~fwang/ ~{!*!*GiIn2;JY#,G?<+TrHh#,G+G+>}WS#,NBHsHgSq~} ============================================================================== From: fwang@hilbert.math.uci.edu Subject: Re: help with MULTIGRID Date: 12 Mar 1999 15:54:57 -0800 Newsgroups: sci.math.num-analysis In article <7ca1ee$cle$1@news.uky.edu>, Jun Zhang wrote: >Feng: > >In your case, you probably do not need multigrid, just a single >grid relaxation will do the job. Exactly. >In fact, when I used a >multigrid and a single grid relaxation to solve a convection >diffusion equation with the Reynolds number (Re) being varied. I >found when Re is small (diffusion like), multigrid is much faster >than single grid; when Re is moderate, say Re = 100 ~ 1000, both >methods converged fast. Multigrid took fewer iterations, but >single grid took less CPU time. When Re is large, both >multigrid and single grid methods converged very slowly. I believe if you use the cross-wind block relaxation method, the single grid one can converge very fast in the high Reynolds number case. I have studied simple cases where the grid is shape-regular and the discretization scheme is monotone, and using piecewise linear elements. In those tests, the more nearly hypobolic the problem is, the faster the convergence. Also, adaptive multigrid method can work very well in these cases, but it can be very complicated to design. >Of course, different multigrid can be designed to make such >problems converge faster, but they are not authentic multigrid >anymore. In fact, it is not uncommon for an nonexpert to >spend one or two years to code a multigrid method for >his/her special applications, I just heard a case like >this. > Agreed. Feng Wang -- ~{KDTB~} http://www.math.uci.edu/~fwang/ ~{!*!*GiIn2;JY#,G?<+TrHh#,G+G+>}WS#,NBHsHgSq~}