Projection method and mesh free
 

I have a problem in my code (diverge)
to solve navier stokes equation, i use mesh free method (uniform) and projection method; in the boundary condition I have Neumann boundary for pressure and for intermediate velocity u^*=2 and u^* =0,I use U^*.\hat{n} = U^{n+1}.\hat{n} but I don't update the velocity in boundary ie i give all time u=2 and 0 is correct???
I don't know what's the problem,
help please :( :( (sorry for my english)

Are you using an implicit method for diffusion in the momentum?

if the physical u at the boundaries is steady you can let the same value but u* = u_n+1 is first order accurate in time (read the paper of Kim and Moin on JCP).
However, that should not cause the divergence of the solution therefore check in the code for some bugs.

I use Crank-Nicolson scheme for convective and diffusive terms

how do you treat the non-linear term in the matrix?

thank you for your reply.
I supposed that the term u_n+1 is transported by u_n

so you are linearizing at first order...but the method should be second order accurate in time..

I suggest:

1) Use CN only for the diffusion and the second order Adams-Bashforth for the convective term
2) Check if you satisfy the continuity constraint in each cell at machine accuracy. You can see that also after 1 time step

thank you I'll try that.
can you give me more information on suggestion 2)

Hi Sir
I have a variable density, to solve the pressure equation I use radial basis function,( its the mesh method that I use)
What is your advice?
Thank you