Crank-Nicolson method and mixed derivatives

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.

• I'm a bit confused -- usually, Crank--Nicolson specifically refers to a time stepping scheme, but these are stationary problems? – Christian Clason Nov 24 '17 at 22:52
• 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. – Charles Nov 25 '17 at 21:15
• @Selig5576, what problem exactly are you experiencing with the mixed derivatives? – Charles Nov 25 '17 at 21:20
• 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! – Simon Nov 26 '17 at 15:40