Dear friends:

Could you plese give me some reference about how to solve the N-s equations in a coupled form.

Generally, i use the segerated solver, that is solve u, solve v, then solve pressure equation. regards

Beam, Richard M. & R. W. Warming, "An Implicit Factored Scheme for the Compressible Navier-Stokes Equations", AIAA Journal, vol 16, no. 4, April, 1978, pp. 393-403.

Briley, W. R. & H. McDonald, "Solution of the Multidimensional Compressible Navier-Stokes Equations by a Generalized Implicit Method", Journal of Computational Physics, vol. 24, pp. 372-397, 1977.

These were widely regarded to be the same idea, same scheme, developed independently at NASA Ames in California (Beam & Warming) and Pratt & Whitney in Massachusetts (Briley & McDonald).

The hard part is that the Jacobian is indefinite (a generalized saddle point problem) which is difficult to precondition. Normally you will want a Schur complement preconditioner, of which there are many varieties. These references are reasonably general, but should give you a good start. Good scalability is still obtainable, for instance (http://59A2.org/files/cheb-scaling.png) shows the results of some experiments I did this spring. The Stokes case uses GMRES preconditioned by a partial solve with an approximate Block LU decomposition where a V-cycle of AMG (ML in this case) is used in the approximate Schur complement.

@article{benzi2005nss, title={{Numerical solution of saddle point problems}}, author={Benzi, M. and Golub, G.H. and Liesen, J.}, journal={Acta Numerica}, volume={14}, pages={1--137}, year={2005}, publisher={Cambridge Univ Press} }

@article{deniet2007tps, title={{Two preconditioners for saddle point problems in fluid flows}}, author={de Niet, AC and Wubs, FW}, journal={Int. J. Numer. Meth. Fluids}, volume={54}, pages={355--377}, year={2007} }

True for a steady-state formulation.

The original references (Beam and Warming, etc) are transient formulations that put 1/delt t on the diagonal so that the coefficient matrix is non-singular. This allows (in principal) solution of the linearized system by non-iterative methods.

Ztdep now has two direct approaches with different tools and difficulties at his disposal.

Try The Integrated Space-Time Finite Volume Method by Philip J. Zwart, Waterloo, 1999 and M. Raw, Robustness of coupled algebraic multigrid for the NS-equations, AIAA Paper 96-0297, 1996

I think it is the solver CFX is using, I implemented it and it is very fast and robust...