# STONE'S STRONGLY IMPLICIT SOLVER

 Register Blogs Members List Search Today's Posts Mark Forums Read

 August 11, 2006, 16:10 STONE'S STRONGLY IMPLICIT SOLVER #1 anybody Guest   Posts: n/a Hi, ich habe eine Frage zum SIP. Bei Wikipedia wird der Algorithmus erklärt (http://en.wikipedia.org/wiki/Stone_method). Leider geht für mich aus der Beschreibung nicht hervor, wie die Matrix N bestimmt wird. Ax = (M-N)x = (LU-N)x = b

 August 12, 2006, 08:10 Re: STONE'S STRONGLY IMPLICIT SOLVER #2 ztdep Guest   Posts: n/a what did you say!

 August 12, 2006, 08:47 Re: STONE'S STRONGLY IMPLICIT SOLVER #3 anybody Guest   Posts: n/a Hi, I don't understand the SIP (http://en.wikipedia.org/wiki/Stone_method). One reason is, that there is no explanation what does N mean. Ax = (M-N)x = (LU-N)x = b If A and b are known where is the reason to go on with an iterative process? The L-U-decompensation leads to the exact result....

 August 14, 2006, 05:27 Re: STONE'S STRONGLY IMPLICIT SOLVER #4 Tom Guest   Posts: n/a "If A and b are known where is the reason to go on with an iterative process? The L-U-decompensation leads to the exact result...." Because A is generally a very large (sparse) matrix whose LU factorization is dense => very large amount of memory to store ; e.g. if you have 1 million grid points then in double precision you would reguire around 7450Gb of RAM just to store the complete LU factoriztion. For a Laplacian type stencil (7 nonzero entries per row in A) then the ILU will only require around 54Mb (and significantly less if the matrix entries are constant)*. Also, in gerneral, the number of floating point operations will be lower for a suitably accelerated iterative scheme with a good initial guess compared to the full LU (assuming you could store it). (*) SIP will actually require the full 54Mb even when the entries of A are constant unlike ILU due to extra interpolations performed in SIP. NOTE: SIP only works for matrices (A) obtained via a discretization of a PDE unlike ILU.

 August 16, 2006, 14:22 Re: STONE'S STRONGLY IMPLICIT SOLVER #5 anyone Guest   Posts: n/a "Because A is generally a very large (sparse) matrix whose LU factorization is dense => very large amount of memory to store ; e.g. if you have 1 million grid points then in double precision you would reguire around 7450Gb of RAM just to store the complete LU factoriztion. For a Laplacian type stencil (7 nonzero entries per row in A) then the ILU will only require around 54Mb (and significantly less if the matrix entries are constant)*. Also, in gerneral, the number of floating point operations will be lower for a suitably accelerated iterative scheme with a good initial guess compared to the full LU (assuming you could store it)." I know that A is a sparse matrix. I thought the L-U algorithm could be used in this way: A*u = r and A=L*U L*q = r --> U*u=q (forward/backward substitution) This would lead to the exact solution u if the right hand side q and A are known. I cannot see why iterations would be necessary... How did you calculate the required mem for 1.000.000 grid points? Thx.

 August 17, 2006, 03:20 Re: STONE'S STRONGLY IMPLICIT SOLVER #6 Karsten Guest   Posts: n/a As you have already written: Ax = (M-N)x = (LU-N)x = b so A=(LU-N) and not A=LU , LU is not the complete LU decomposition, meaning it has only nonzero elements where A has nonzero elements. It is sparse, while the full LU decomposition wouldn't be so sparse. The other elements are stored in the matrix N which of course could be calculated by N=LU-A. This decomposition is used in iterative methods: Mx = Nx + b can be solved iteratively by: M x_i+1 = N x_i + b or x_i+1 = M^(-1) N x_i + M^(-1)b This is useful If M can easily be inverted, and M^(-1)N has specific properties (abs of largest eigenvalue <1).

 August 17, 2006, 06:19 Re: STONE'S STRONGLY IMPLICIT SOLVER #7 Tom Guest   Posts: n/a "How did you calculate the required mem for 1.000.000 grid points?" It's what would be required to store a full 10^6 x 10^6 matrix which you would have to consider doing if you formed the full LU factorization. As an exercise write down the finite difference equations for the Laplace equation on a square with 10^2 and 25^2 points and form the LU factorization of the matrix and see how many zeroes there are and compare this to the original matrix which has just 5 bands.

 August 18, 2006, 09:40 Re: STONE'S STRONGLY IMPLICIT SOLVER #8 anyone Guest   Posts: n/a Could you give an explicit example for N. I read several papers about the L-U-decompensation and I got mixed up with it. I got the impression the word L-U-decompensation is interpreted in different ways. [2.0 1.0 0.0 0.0 0.0 0.0 0.0]*****[1.0 0.0 0.0 0.0 0.0 0.0 0.0]*****[2.0 1.0 0.0 0.0 0.0 0.0 0.0] [1.0 2.0 1.0 0.0 0.0 0.0 0.0]***** [0.5 1.0 0.0 0.0 0.0 0.0 0.0]***** [0.0 1.5 1.0 0.0 0.0 0.0 0.0] [0.0 1.0 2.0 1.0 0.0 0.0 0.0]*****[0.0 0.667 1 0.0 0.0 0.0 0.0]*****[0.0 0.0 1.3 1.0 0.0 0.0 0.0] [0.0 0.0 1.0 2.0 1.0 0.0 0.0]**=**[0.0 0.0 0.749 1 0.0 0.0 0.0]*****[0.0 0.0 0.0 1.25 1 0.0 0.0] [0.0 0.0 0.0 1.0 2.0 1.0 0.0]*****[0.0 0.0 0.0 0.8 1.0 0.0 0.0]*****[0.0 0.0 0.0 0.0 1.2 1.0 0.0] [0.0 0.0 0.0 0.0 1.0 2.0 1.0]*****[0.0 0.0 0.0 0.0 0.833 1 0.0]*****[0.0 0.0 0.0 0.0 0.0 1.166 1] [0.0 0.0 0.0 0.0 0.0 1.0 2.0]*****[0.0 0.0 0.0 0.0 0.0 0.857 1]*****[0.0 0.0 0.0 0.0 0.0 0.0 1.14] *****A*****=**********L*********************U One paper calls this a L-U-decompensation....which would not be right with your definition! I think I understood, why it makes sense to solve the system of equations iteratively. Using SIPSOL for non-linear equation systems means that A is not constant because the coefficients include mass-fluxes (rho*u)!

 August 18, 2006, 10:01 Re: STONE'S STRONGLY IMPLICIT SOLVER #9 Tom Guest   Posts: n/a The LU in SIP is not the LU decomposition it is the Incomplete LU (or ILU see the book by Peric & Ferziger for an explanation). The full LU has N=0 but because ILU is not the complete factorization there is an error (N = LU-A is nonzero).

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules

 Similar Threads Thread Thread Starter Forum Replies Last Post sek OpenFOAM Running, Solving & CFD 38 July 11, 2015 04:59 felixrieper OpenFOAM Running, Solving & CFD 0 May 18, 2006 07:19 Henrik StrÃ¶m FLUENT 1 October 29, 2005 03:57 youngan CFX 0 July 1, 2003 23:32 N.SUBRAMANIAN Main CFD Forum 6 August 20, 2000 22:24

All times are GMT -4. The time now is 21:15.