What does the Navier-Stokes equation look like after time discretization?

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The discussion centers on the time discretization of the Navier-Stokes equation as presented in the Gerris flow solver paper by Prof. S. Popinet. The author expresses confusion over the use of different time intervals, specifically the terms un+0.5 and un+1, and how they relate to density and velocity calculations. It is clarified that the time step projection method is employed, which computes an intermediate velocity before updating it to ensure divergence-free conditions. The author questions the rationale behind using density at n+0.5 without an advection equation, suggesting that density terms at n and n+0.5 should be equivalent. Overall, the discussion seeks clarity on the finite difference scheme and its implications for solving the Navier-Stokes equation.
Kukkat
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Hi,

I know the general form of the Navier Stokes Equation as follows.
upload_2016-12-2_12-30-26.png


I am following a software paper of "Gerris flow solver written by Prof. S.Popinet"
[Link:http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.374.5979&rep=rep1&type=pdf]
and he mentions after time discretization he ends with the following equation:
upload_2016-12-2_12-31-35.png

where n-1 is the previous time step, n+1 is the next time step and n+0.5 is mid time for the present time step.

Solving equation implicitly/ explicitly in time means solving for next time data however in the equation there are rather two unknowns un+0.5 and
un+1.

Not sure why he uses different terms at different time intervals. Density at n+0.5, velocity at n, n-1, n+0.5 etc..

Can anyone point me or explain me how he arrives at this specific sort of discretized equation.
 
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The link to the paper doesn't work.
 
I'm not familiar with this particular finite difference scheme, but presumably un+0.5 is already know when you are calculating un+1
 
Sorry for the link.
http://www.sciencedirect.com/science/article/pii/S002199910900240X

@Chestermiller guess that's true. It is being solved by the time step projection method which means an intermediate velocity is computed and later updated to a divergence free velocity by solving the laplace of pressure term as in the mentioned paper.

The notation is a bit strange for me as he uses density at time n+0.5 without solving any advection equation. From what I see density terms at n and n+0.5 should be the same.
 
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