0
$\begingroup$

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.

| cite | improve this question | |
$\endgroup$
  • 3
    $\begingroup$ I'm a bit confused -- usually, Crank--Nicolson specifically refers to a time stepping scheme, but these are stationary problems? $\endgroup$ – Christian Clason Nov 24 '17 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 Nov 25 '17 at 21:15
  • $\begingroup$ @Selig5576, what problem exactly are you experiencing with the mixed derivatives? $\endgroup$ – Charles Nov 25 '17 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 Nov 26 '17 at 15:40
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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 Nov 27 '17 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 Nov 27 '17 at 22:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.