Multigrid vs. non-stationary iterative solver

mb.pejvak · October 18, 2019, 11:36

As I know multigrid since its concept and the idea behind it can converge faster in comparison to stationary methods. I also ran some cases in openFoam in similar conditions, but the iterative solver changed (non-stationary and multigrid methods). to summarize

iterative solver & Time of execution (sec)
Multigrid & 28118
PCG(GAMG) & 63881
PCG(DIC) & 86274

As you can see, multigrid almost 3 times faster than CG with DIC as preconditioner and 60% faster CG with multigrid as preconditioner.
Now my question is if it can be generalized that multigrid is the fastest method in solving problem related to CFD

LuckyTran · October 18, 2019, 16:45

It is an okay generalization but there are limits.

Multigrid methods accelerate convergence when there is a scale disparity. This is easiest to appreciate when solving steady state problems on a very fine grid because the discretized system is very good at reducing small wavelength errors but not so good at reducing large wavelength errors.

If you solve transient problems with a very small time-step (or analogously, use a steady solver with implicit under-relaxation with a very small under-relaxation factor), the large wavelength errors are not the ones stalling convergence anymore and there is not much benefit to acceleration.

You also have to consider exactly which multigrid cycle is being implemented (V vs W vs VW vs WW, etc). A W or W+ cycle (as a hyperbole, consider a 24W cycle) can easily exceed the cost of simply doing another iteration. In other words, by stupidly using a multigrid method, you can also solve problems much slower.

mb.pejvak · October 19, 2019, 02:56

Thanks LuckyTran for the answer.
As I know, by Fourier analysis of error in iterative method (e.g. Jacobi), the maximum eigenvalue (corresponds to max frequency of error) for Jacobi method is
$\lambda_{max} = cos(\pi h) \approx1 - 1/2(\pi h)^2$
So if there is wide range of spacial step size (h), it results in wide range of frequency (eigenvalue) in the error, and it may leads to stiff problem. in this case multigrid can be beneficial.
But about the impact of temporal step size, how the eigenvalue of the matrix change by the combination of spacial and temporal step size. introducing any reference would be appreciated.

October 18, 2019, 11:36	Multigrid vs. non-stationary iterative solver	#1
mb.pejvak Senior Member Mehdi Baba Mehdi Join Date: Jan 2011 Posts: 158 Rep Power: 15	As I know multigrid since its concept and the idea behind it can converge faster in comparison to stationary methods. I also ran some cases in openFoam in similar conditions, but the iterative solver changed (non-stationary and multigrid methods). to summarize iterative solver & Time of execution (sec) Multigrid & 28118 PCG(GAMG) & 63881 PCG(DIC) & 86274 As you can see, multigrid almost 3 times faster than CG with DIC as preconditioner and 60% faster CG with multigrid as preconditioner. Now my question is if it can be generalized that multigrid is the fastest method in solving problem related to CFD Last edited by mb.pejvak; October 19, 2019 at 04:49.

October 18, 2019, 16:45		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,675 Rep Power: 66	It is an okay generalization but there are limits. Multigrid methods accelerate convergence when there is a scale disparity. This is easiest to appreciate when solving steady state problems on a very fine grid because the discretized system is very good at reducing small wavelength errors but not so good at reducing large wavelength errors. If you solve transient problems with a very small time-step (or analogously, use a steady solver with implicit under-relaxation with a very small under-relaxation factor), the large wavelength errors are not the ones stalling convergence anymore and there is not much benefit to acceleration. You also have to consider exactly which multigrid cycle is being implemented (V vs W vs VW vs WW, etc). A W or W+ cycle (as a hyperbole, consider a 24W cycle) can easily exceed the cost of simply doing another iteration. In other words, by stupidly using a multigrid method, you can also solve problems much slower. mb.pejvak likes this.

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October 19, 2019, 02:56		#3
mb.pejvak Senior Member Mehdi Baba Mehdi Join Date: Jan 2011 Posts: 158 Rep Power: 15	Thanks LuckyTran for the answer. As I know, by Fourier analysis of error in iterative method (e.g. Jacobi), the maximum eigenvalue (corresponds to max frequency of error) for Jacobi method is $\lambda_{max} = cos(\pi h) \approx1 - 1/2(\pi h)^2$ So if there is wide range of spacial step size (h), it results in wide range of frequency (eigenvalue) in the error, and it may leads to stiff problem. in this case multigrid can be beneficial. But about the impact of temporal step size, how the eigenvalue of the matrix change by the combination of spacial and temporal step size. introducing any reference would be appreciated.