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Coupled Solver Include Y-velocity terms in X-momentum equation? |
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#21 | |
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Raphael
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#22 | |
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The unexpected was that SGS basically always works, in a way or another, and the difference with GMRES+ILU0 is not order of magnitudes, but O(1), maybe 10 sometimes. Yet SGS is really faster, so the overall time is down to comparable in most cases. It is so for both convection and diffusion dominated problems, and the main explanation I found is that, no matter what, you are still solving a sytem linearized at a given pseudo-time/outer iteration. If it is, say, the first outer iteration, there is little to gain from oversolving it. And it is the same for most of the initial phase. See, for example, here: https://aisberg.unibg.it/retrieve/ha...ntinuation.pdf http://www.ii.uib.no/~trond/publicat...rs/Inexact.pdf https://users.wpi.edu/~walker/Papers...1996,16-32.pdf In practice, you mostly need just 1-2 orders in the residual reductions per outer iteration, and SGS can actually give them to you. SGS will fail in going any further, but that is not required in most cases. |
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#23 | ||||||
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Raphael
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I am still trying to understand how this works. you multiply block Aii by the inverse of Aii, I assume you also multiply Aij (across, not down) by the inverse of Aii for all Aij=/=0? It is sort of like combining U V and P into one variable and performing Guass seidel on that? |
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#24 | |
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Raphael
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Ok, I guess i mistook and didnt realize SGS is different than GS. From this reference https://math.stackexchange.com/quest...idel-iteration it seems like SGS inverts the (D+L) matrix to get x_k+1/2, and then uses x_k+1/2 to get x_k by inverting (D+U) matrix. It is sort of like assuming that the east and north values are fixed and solving for east/west, and then reversing that assumption. Because it uses a matrix inverse, i expect: (1) That low frequency errors will be dissipated fast (unlike regular GS) (2) That it is much less expensive to get the inverse of (D+L) and (D+U) than (D+L+U). I suppose (D+L) is like an advection only problem i.e. it is parabolic not elliptic, so a sweep in the right direction will solve it even with regular GS. |
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#25 | |
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YES. Not sure for other methods, but for GS it is EXACTLY THAT. |
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#26 | |
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#27 |
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Raphael
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Right. It is explicit though, rather than implicit.
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#28 | |
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Raphael
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But in that case, i would expect it to be terrible at dampening out the low frequency errors. |
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#29 |
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#30 |
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Raphael
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Thanks again for all your help.
I am looking into Rhie Chow and collocated grid. I have the CFD textbook by Ferziger, and it describes a term like Rhie Chow that is proportional to the third derivative of the pressure. It also appears in a similar form as Rhie Chow but is not labeled as Rhie Chow, so I am not quite sure if it is the same thing or not. For the purpose of this post, I am going to assume it is the same thing, but kindly please let me know if it is not. As far as i can tell, the basic idea behind the Rhie Chow method is to solve the momentum equations using the same CVs as the continuity and other scalars, but add corrections to momentum and/or continuity equations for consistency and energy conservation and to prevent oscillating pressure fields. That said I have a few questions, if you dont mind: (1) At this point, I dont know how the cell-centered pressure gradient in the momentum equations is computed-- is it simply computed using central difference [(P_E-P_W)/(2*dx)], which results in coefficients of P_P = 0 (since no P_P term appears in the expression above), or is there a Rhie chow correction term (and if so, how is it defined)? (2) From Ferziger, it seems like there is a Rhie Chow correction in the momentum equation at the eastern face (which is probably used for continuity), but i dont know if there is any correction at the cell center? (3) The velocity corrections are applied to the faces, seemingly without a Rhie Chow correction. The only difference is that the A_p coefficients are computed by interpolation onto the faces rather than getting the A_p coefficient using the values of face velocities (as in a staggered grid). I assume then that the cell velocity corrections are applied by averaging the face corrections. |
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#31 | |
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Arjun
Join Date: Mar 2009
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This is indeed true. I actually first came to know about it when I was starting off my career and we had to use a code written at Tohoku university. They use this for high speed simulation. |
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coupled solver, linearize, segregated solver |
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