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I am curious if anyone had literature references or knowledge on how to apply the Crank-Nicolson (with approximate factorization) to the $$ \nabla \cdot (\nu (\nabla \mathbf{u} + \left(\nabla \mathbf{u}\right)^{T})) $$

What I am unsure about is how to handle the mixed derivative. Currently I am using Crank-Nicolson for

$$ \nu_{Laminar}\nabla^{2} \mathbf{u} $$

which is straight forward as I can solve it dimension-by-dimension.

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    $\begingroup$ I'm a bit confused -- usually, Crank--Nicolson specifically refers to a time stepping scheme, but these are stationary problems? $\endgroup$ Commented Nov 24, 2017 at 22:52
  • $\begingroup$ I agree with @ChristianClason. That said, I think that the OP only wrote the terms of interest, but it should be clarified in the context. $\endgroup$
    – Charles
    Commented Nov 25, 2017 at 21:15
  • $\begingroup$ @Selig5576, what problem exactly are you experiencing with the mixed derivatives? $\endgroup$
    – Charles
    Commented Nov 25, 2017 at 21:20
  • $\begingroup$ Thanks for the replies. When I talk about Crank-Nicholson with approximate factorization, I am talking about applying Crank-Nicholson, but I solve the u_xx, u_yy, u_zz, individually (with TDMA). What I confused by is since I am using dimensional splitting for my viscous term, how can I calculate the mixed derivative since I am solving directionally. If it is of any relevance, I have developed a collocated incompressible NS FVM based solver and am interested in studying turbulence with my solver. Thanks! $\endgroup$
    – Simon
    Commented Nov 26, 2017 at 15:40

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I suggest you just use CG method, and write a routine that computes Ax, where Ax is your momentum diffusion term. Then just use a matrix free implementation of CG. The only requirement is that A is symmetric positive definite, which I think it is for this case.

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  • $\begingroup$ So it is not possible to use an ADI type method for the full viscous stress tensor due to the mixed derivatives? Thanks for the suggestion! $\endgroup$
    – Simon
    Commented Nov 27, 2017 at 19:09
  • $\begingroup$ I'm not certain that it is not possible to use an ADI type method, but I'm fairly sure that CG method can be used. $\endgroup$
    – Charles
    Commented Nov 27, 2017 at 22:04

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