[CFD] Gauss-Seidel Method in CFD 

For a thirty minute review, this is pretty good. I think a followup review could be helpful that connects the Gauss-Seidel method to the splitting where Ax=b is approximated assuming A=L+D+U where L is strictly lower triangular, D is a diagonal matrix and U is strictly upper triangular, then we can approximate the inverse of A by subdividing the matrix such that we have (L+D)x^k+1 + Ux^k = b, or alternatively Lx^k+(D+U)x^k+1 = b. The renumbering of the equations has the effect of moving the coefficients around between the L and U matrices. The triangular matrix has a trivial inverse (which is what is implemented by the sweeping approach described in this video). It is important to also note that *when* the coefficients come from an upwind scheme, then it is possible to number the equations such that there are many more zeros in either the L or U sub-matrices. This will result in a much more effective iterative scheme, whereas for coefficients from an elliptic system such as the heat equation give equal weight to the L and U parts. This usually means that the effectiveness of the Gauss-Seidel method will be more effective for upwind dominant systems often found in higher speed compressible flows compared to lower speed diffusion dominant flows. For diffusion dominant problems the Gauss-Seidel method will quickly stall and must be combined with a multigrid or Krylov subspace method to actually be useful.

Great! Before my exam it was mindblowing.

Wow, this video is such a great explanation of the Gauss-Seidel algorithm! I really appreciate how it was broken down and explained using a basic example with limited math. As someone who is new to CFD, this was the perfect level of detail for me to understand the concept. Thank you for sharing!

Thank you! I was just thinking about how little I knew about matrix methods

This is gold. Much obliged good sir

These are very helpful! Thank you for these great videos and excited for conjugate gradient! I always see the preconditioned bi-conjugate gradients in openfoam and have zero clue what they do lol

Looking forward to the conjugate gradient and multigrid methods!

Thank you for your great work! I would love to see a Video on the ANSYS Fluent Adjoint solver, for example on a lift/drag optimization. Keep it up!

Amazing video, I really enjoyed it, thanks for delivering this high quality material

Yes, finally this is the video I have been waiting for from you Dr Aidan. Thank you, I can finally use the GS method properly.

Great video. Thank you

Wow, that's a really basic algorithm! Very interested to review how these actually function! And what that actually means.

thank you 🙏🏻 new video here is always a good news, merry xmas, Aidan 🎄🎉❄️

It was another very usefull talk :) More deeper understanding is gained :)

Great explanation!!! Can you please come with an explanation for for solving non liner problems ?

Amazing video! Would be awesome if you did other linear solver videos like DADI or GMRES. :)

I was listening to your talk with Jousef Murad and found it cool that you were would like to know how SpaceX does their supersonic and hypersonic computations. Are you going to make a video on compressible solvers at some point? That would be great! The landscape of compressible solvers is a complex one to navigate (density based vs pressure based, riemann solvers, etc.)

While fairly obvious, but not explicitly mentioned herein, the convergence of this method will also depend on the initial guesses.

nice lecture... please make similar ones for conjugate gradient and multi-grid methods

Thanks for a great series of videos! How is the Gauss-Seidel method (and other general solvers such as multi-grid) related to the SIMPLE and PISO you talked about early in this series? Are they just different ways to solve the NS equations? When do you prefer which methods?

Thanks! Do you use beamer + tikz/inkscape to create these beautiful slides?

I wanted to ask in the Q&A of the webpage but I think there isn't. Is it going to be a Christmas sale or something like that? I from LATAM and the courses are a bit expensive ( by no means I saying they wouldn't be worth the price just that the currency convertion :c )

I really want to get into CFD career. Can you make a beginner friendly video lectures. I have enrolled into your fundamental but most of stuff is hard to grasp. I have to continuously look up scientific notations. I am fairly good with calculus 1, can you suggest prerequisite to the fundamentals course? Thanks

Hello. Are you going to do some videos about chemically reacting flow in CFD? Or this is not your area of interest?

I think your presentation would benefit from pointing out, though notation, when a variable is assumed to be know from a previous iteration. E.g. your Eqn 11 is not the formula for Gauss-Seidel, it is missing appropriate notation and terms you mentioned but choose to ignore. You couldn’t code the method based on your talk. (This type of clear differentiation of variables through notation would also have really benefited your talk on the SIMPLE method.) Also, typically, algorithms are not deemed to be converged when the difference from iteration to iteration is small, but when some norm of the residual r:=b-Ax is also small. If only the differences are considered the algorithm may have stagnated without convergence. GS is not used alone, it is either used as a smoother in Multigrid or in some modified form as a preconditioner to a Krylov method; on its own the convergence rate is too slow. I too strongly recommend Youssef Saad’s freely available book on iterative methods to anyone interested in further exploring the linear algebra relevant to CFD. (Google “yousef saad iterative methods” for the pdf.)