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Will a multigrid method damp out the high frequency physical wave? 

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September 12, 2009, 11:03 
Will a multigrid method damp out the high frequency physical wave?

#1 
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Edison
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I am solving timedependent problem, which contains physical waves with different frequencies. The NS equations I use contain a Poisson's equation. I use iterative method to solve it, but I find the convergence speed is very low.
Hence, I hope to use a multigrid method to accelerate the convergence. However, I am afraid that the high frequency physical waves cannot be represented in a coarse grid and my simulation target will be damped out. Will this happen? By the way, I feel rather confused about the concept of "low frequency errors" in multigrid method. Is it the same thing as the physical waves with low frequency I need to simulate? Thank you in advance. 

September 12, 2009, 12:47 

#2  
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Quote:
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No. The coarser grids are just used to get an error estimate that is then "prolonged" back to the fine grids. In the end you still have a solution on your fine grid that includes the higher resolution information. Quote:
Not really; lots of iterative methods (eg. point GaussSeidel, SSOR) are good at getting rid of high frequency error, but take many iterations to smooth out the lowfrequencies. By going to a coarse grid, what used to be low frequency errors on the fine grid become higher frequency (which the method handles well), so convergence speeds up. Quote:
HtH, Josh 

September 21, 2009, 00:42 

#3 
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sato
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Hi,
I have encountered the same problems as you. Do you mean that your MG doesn't work well? I have experiences that at first I checked up MG's procedure by programming 1D problem and then challenged 3D one which was my aim. http://www.geocities.jp/viola08129/ 

September 29, 2009, 00:25 

#4 
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Edison
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Thank you all very much.
I have programmed the multigrid method. However, I really find the wave amplitude is 10% smaller by multigrid method than only using a fine grid. Is this a normal result of multigrid method? By the way, when solving the Poisson’s equation, I didn’t use the restriction operatior on the source term. Is it necessary? Thank you. Last edited by hadesmajesty; September 29, 2009 at 23:33. 

September 29, 2009, 10:16 
restriction operatior etc.

#5 
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sato
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Hi,hadesmajesty
[QUOTE=hadesmajesty;230777]Thank you all very much. I have programmed the multigrid method. However, I really find the wave amplitude is 90% smaller by multigrid method than only using a fine grid. Is this a normal result of multigrid method? It isn't a normal result, I think. By the way, when solving the Poisson’s equation, I didn’t use the restriction operatior on the source term. Is it necessary? Necessary for rsiduals(source term?). V cycle procedure I used with max. grid level 3 for linear equation system(1D problem) Aw=G is as follows; w=0 ! initial solutions f[3]=G ! initial residuals at max. grid level=3 do i=1,niter ! n in [n]:grid level f[2]=R[2,3]*f[3] f[1]=R[1,2]*f[2] u[3]=0 ! initial residual at g.l.=3 u[3]=relax(A[3]*u[3]=f[3]) ! relaxation for residual eq. at g.l.=3 f[2]=R[2,3]*(f[3]A[3]*u[3]) ! restriction for residuals f u[2]=0 u[2]=relax(A[2]*u[2]=f[2]) f[1]=R[1,2]*(f[2]A[2]*u[2]) u[1]=inv(A[1])*f[1] ! exact solution at g.l.=1 u[2]=u[2]+I[2,1]*u[1] ! prolongation and correction u[3]=u[3]+I[3,2]*u[2] f[3]=f[3]A[3]*u[3] w=w+u[3] ! correction for solution end do Thank you. 

October 2, 2009, 14:23 

#6 
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Josh
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Do you run a sweep (or a couple) of your solver one last time on the fine grid to kill off the highfrequency errors? You should be converging to the same solution, multigrid just gets you there with less work (hopefully).


October 7, 2009, 04:14 

#7 
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Edison
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Thank you very much, jstults and trout. I think I have found what’s wrong.
To solve Ax=b. Last time I used the iteration scheme like, x(n)=G*x(n1)+b, where G and b are fixed in all iterations, and just transferred the unknown x and the source term b between the coarse and fine grids. Now I used the iteration scheme: Residual=(A*x(n1)b) dx=G’* Residual x(n)= x(n1)+dx I transfer the residual (A*x(n1)b) between the two grids. The multigrid cycle is that: first sweep two times on the fine grid, then transfer and sweep one time on the coarse grid, and then go back to fine grid. This time I find the result is the same as just using fine grid. However, the above algorithm has larger computational cost, because I have to do matrix multiplication for two times : first A*x(n1) for the residual and second G’* Residual for dx. Therefore, the final computational times are as below: 1. directly solve x only on fine grid: 4.46s.(my old method) 2. Solve residual and then dx only on fine grid: 6.73s. 3. Solve residual and then dx by multigrid: 4.61. So, compare the time used by 1. and 3. The multigrid method cannot improve anything. Is my current algorithm correct? Or is there any place to improve? Thanks again. 

October 7, 2009, 11:58 

#8  
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Josh
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Quote:
The early slides are also good, they talk to your original questions about the low/high frequency errors. 

October 13, 2009, 00:26 

#9 
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Edison
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Thanks a lot.
I follow the book of William Briggs and use a twolevel coarse grid correction method. The solution is correct, but the convergence speed has not been improved or even lower. The Poisson's equation I solve is a*d^2u/dx^2+d^2u/dy^2=f, in which 0<a<<1. Hence, I use GaussSeidel line iteration by solving variables in y direction at the same time. In the coarse grid correction step, according the book of William Briggs, I just coarsen the grid in y direction. For each line, I calculate the residual, then the error and update the new value. After that I go to next line and repeat the above procedure again. I am not sure how to implement the coarse grid correction method with the GaussSeidel line iteration. Is my algorithm correct? Thank you very much. 

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