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 Amith April 5, 2002 02:16

Multigrid methods

Hi,

I was curious to know why Multi grid methods cannot be applied to hyperbolic equations and is applicable only for elliptic equations.

Regards, Amith

 andy April 5, 2002 08:38

Re: Multigrid methods

Multigrid methods do get applied to hyperbolic equations and they often accelerate the convergence rate to some degree. However, they almost never approach the performance achievable when used with elliptic equations.

The performance of a multigrid method for a particular equation or, more usually in CFD, set of equations is dependent on the details of the various components in the multigrid scheme. That is, the prolongation method, the restriction method, the cycling strategy, the coupling strategy, the smoother, etc... A particular scheme can work extremely well for one set of equations and simply fail for another set. The reasons for good performance or bad are going to vary from scheme to scheme and set of equations to set of equations.

The pat answer to your question is the lack of smoothness in the solutions to hyperbolic equations compared to elliptic equations. See any multigrid book for a description why this property is important for a multigrid scheme. Nonetheless, there are usually things one can do in practice to ease the situation and get some acceleration of convergence.

 Frederic Felten April 5, 2002 16:43

Re: Multigrid methods

Hi there,

For elliptic problems (like a pressure poisson equation), the solutions is driven by the boundary conditions (this is not the case for hyperbolic problems). The regular iterative methods are slow to propagate the information troughout the domain especially for the low frequencies. The multigrid method is then a fast way of get rid of this slow convergence process.

Sincerely,

frederic Felten.

 Neale April 5, 2002 23:50

Re: Multigrid methods

You could use a multigrid method on a hyperbolic equation, but it probably would not help all that much. Multigrid tends to do well on elliptic problems because they are non-local. In particular it's really good at smoothing out long wavelength errors over the entire domain.

Hyperbolic problems tend to behave like wave propagation, which can only propagate at a specific speed. Wave propagation is not non-local like diffusion, in that influence effects are only felt after a specific amount of time. Diffusion acts everywhere all at once. As a result, standard iterative techniques, such as point Jacobi, or point Guass-Siedel, or ILU, etc... without multigrid, will tend to do just as well as multigrid on pure hyperbolic problems.

Neale.

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