# (Approximate) Incremental Projection Method for Navier-Stokes equations

I am trying to implement an incremental projection method for the 2D incompressible Navier-Stokes. The type of projection method I am trying is

$$\frac{u^{*} - u^{n}}{dt} = - \nabla p^{n} - u \cdot \nabla u + \nu \nabla^{2} u\\ u^{*}|_{\partial D} = u^{n+1}|_{\partial D}\\ \nabla^{2} \phi^{n+1} = \frac{\rho}{dt} \nabla \cdot u^{*}\\ \nabla \phi^{n+1}|_{\partial D} = \frac{1}{dt} \left(u^{*} - u^{n+1}\right)\\ \frac{u^{n+1} - u^{*}}{dt} = - \nabla \phi^{n+1}\\ p^{n+1} = p^{n} + \phi^{n+1} (5)$$

In terms of the velocity-pressure arrangement I am using a collocated formulation using the Rhie-Chow interpolation to have stable coupling. I however have run into a problem in terms of the practical implementation of the pressure update. The way I update pressure is as follows

Pnew(1:nx,1:ny) = P(1:nx,1:ny) + Phi(1:nx,1:ny)
P(1:nx,1:ny) = Pnew(1:nx,1:ny)


I am testing my solver on the lid driven cavity. My velocity profile is correct, however my pressure profile is completely wrong as it contains wiggles. Here is a picture: I have tested my solver on a pressure-free projection method(where there is no $\nabla p^{n}$ in the intermediate veloctiy step and it works fine.

EDIT: (1/17/18)

I forgot to mention but I am using 2nd-order central discretization for both convection and diffusion. My integration technique is currently forward Euler for debugging purposes.

• There are so many reasons why this could be wrong. There are also many reasons why it could be right -- have you investigated whether the pressure solution you get actually converges, and then whether it converges to the correct pressure, and then whether it converges at the right rate? – Wolfgang Bangerth Jan 17 '18 at 20:54
• The pressure profile in my incremental projection method I know to be wrong because it does not agree with the results of my pressure-free projection method in which I have validated with data. With that said, I do find it strange that while the pressure profile has wiggles the velocity profile is correct. In terms of the rate of convergence, in time I am getting O(h^2,dt) which corresponds to the discretization methods I am using. I have edited my post to contain more details on my numerical schemes. – Simon Jan 17 '18 at 22:05
• Bump. Any thoughts? – Simon Jan 23 '18 at 12:14