Hi --

Could someone tell me what the exact disadvantages of Crank Nicholson scheme are (when used to solve the Navier Stokes Equations)? I know that there exists a CFL condition when stability is analyzed in the max norm, but there is no restriction when analyzed in the 2-norm.

I also know that for more than 1-D Crank Nicholson may be too expensive to invert (hence the use of ADI, etc).

What do you think are the main reasons that people prefer not to use Crank Nicholson (or, maybe, prefer to use it)?

I prefer use the Crank-Nicholson because it is second order in time and semi-implicit.

Usaully, to convert the resulting matrix, I use multigrid method which is fast.

Crank Nicholson is neutrally stable scheme. It does not have any algorithmic damping to diffuse any physical or numerical oscilation. If you look at the amplification plot of CN, it has poor performance for high modes. Yes, it is second order accurate but is oscillatory for rough problems. In problems with mesh motion (FSI) CN has poor performance. There is a solution to fix the problem. If you have to use 2nd order in time, thn you can select the modes which are more important for you (Greg Hulbert, Ken Johnson; CMAME 2004 or 2003). They have used their scheme on LES.

Crank Nicholson is most frequently used for discretization of diffusive terms in N-S equations.