# approximate factorisation

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

 November 2, 2005, 20:04 approximate factorisation #1 dist Guest   Posts: n/a can someone please tell me what approximate factorisation technique is? thankyou.

 November 3, 2005, 05:52 Re: approximate factorisation #2 Tom Guest   Posts: n/a It's a way of splitting the matrix. For example for a full Lower/Upper factorization the matrix A is exactly factorized as A = LU while for an approximate factorization such as ILU you would have A = LU + N. Here N is the remainder and the basic idea behind approximate factorization is to make N as small as possible.

 November 3, 2005, 17:08 Re: approximate factorisation #3 dist Guest   Posts: n/a ok, i suppose that makes sence. but why do we use it? and how to we do about ensuring the n matrix is small? thank you for your answer by the way.

 November 3, 2005, 17:48 Re: approximate factorisation #4 ag Guest   Posts: n/a Consider a 2D system where the implicit difference equation is of the form (I + A + B)*Q = RHS The matrix I + A + B will in general be pentadiagonal and is more difficult to solve than the classic tridiagonal matrix. Approximate factorization makes the following substitution: Let (I + A)*Q1 = RHS and solve for Q1, then (I + B)*Q = Q1 to solve for the Q unknowns in the original equation. This factorization yields two systems which are each tridiagonal. The problem is the splitting error, which is found by multiplying out the two pieces: (I + A)*Q1 = (I + A)*(I + B)*Q = RHS or (I + A + B + AB)*Q = RHS The quantity AB*Q is the splitting error, and will in general be a function of time step and cell size and the gradients of the flowfield. In two dimensions AF works reasonably well, but in 3D the splitting error will be O(dt**3) and can render an implicit scheme pointless unless measures are taken to deal with the splitting error. One such approach is to use the AF scheme within a global Newton iteration, such the the residual is driven to zero by the Newton and the AF is just used to get the approximate solution needed for the inner iteration.

 November 3, 2005, 18:12 Re: approximate factorisation #5 ag Guest   Posts: n/a Sorry, that should be O(1/dt**3). It's been a long afternoon.

 November 4, 2005, 07:00 Re: approximate factorisation #6 Tom Guest   Posts: n/a You use it when you can't perform the full inversion efficiently; i.e. the full LU factorization of a sparse matrix A is usually quite dense and so for a large matrix would take a vast amount of memory to store. With my example of ILU you assume that L and U only have non zero entries in the same locations as the original matrix A. Multiplying out LU you then equate the terms to those of A (the remaining terms are N). The main reason for the use of approximate factorizations is that they can be used as postconditioners in other iterative solvers to help speed up convergence (LU is approximately equal to A but is relatively simple to invert).

 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 vahidmech Main CFD Forum 0 December 15, 2009 15:08 Meena Main CFD Forum 1 February 9, 2007 11:04 sebed Main CFD Forum 0 October 19, 2005 18:18 Greg Perkins Main CFD Forum 0 February 12, 2003 19:43 rahul Main CFD Forum 3 June 5, 2002 23:32

All times are GMT -4. The time now is 18:35.