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April 11, 2008, 21:43 
What is PreConditioning?

#1 
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Can anybody tell me what is meant by preconditioning in a cfd simulation?
Cheerios Rif 

April 12, 2008, 11:08 
Re: What is PreConditioning?

#2 
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It is kind of restructuring linear equations systems before solving them. This leads to an quicker solvable equation system. Regards.


April 12, 2008, 14:37 
Re: What is PreConditioning?

#3 
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You will encounter mainly two types of preconditioning: 1.  at the PDE level (and its discretization) 2.  linear system level
1. When simulating (steadystate or unsteady with dualtime formulation) flows using a densitybased (compressible) solver, in the low Mach# regimes (typically below Mach 0.3), the solution algorithm will require "lowMach preconditioning". This is basically a timederivative preconditioning, which will modify the way the solution evolves in pseudotime towards convergence (or at each time step, for unsteady). The whole idea is to produce a different (!) hyperbolic PDE, but which will converge to the same steadystate (or same quasi steadystate, when unsteady), with wavespeeds that are closer in value, thus having much better convergence characteristics. An additional benefit of lowMach preconditioning is that the numerical dissipation of the preconditioned scheme is in a sense .. optimal (thus accuracy improves also, besides convergence). The "modified" PDE will look like: (Gamma/dt+dRes/dw)*dw =  Res(w_n) (where Gamma is the timederivative preconditioning matrix, w is the "work" variable (p,V,T), Q is the conservative variable (rho, rhoV, rhoE)) The nonpreconditioned PDE is just: (dQ/dw / dt+dRes/dw)*dw = Res(w_n) Thus [Gamma] preconditioning matrix replaces the [dQ/dw] Jacobian. 2. When solving the linear system, if the system is illconditioned (large condition number) or the matrix is difficult to invert, one would like to replace the problem of solving using the original matrix, with one of (more) solves with a simpler matrix, call it preconditiner, which is a matrix with much better conditioning (and preserves the same "characteristics" of the original matrix > eigenvalues sign, etc.) and is easier to invert. Of coarse, the final result should satisfy the original linear system of equations. Thus instead of solving A.x + b = 0, you could think of solving in .. correctionform: P.dx = (A.x + b) where P is the preconditioning matrix, and the update x = x + dx, etc. I gave you a glimpse of the real deal, but there are many papers out there on the subject. John C.S. 

April 13, 2008, 14:34 
Re: What is PreConditioning?

#4 
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any books in this subject or background specially when it comes to numerical analysis with fortran


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