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

Hi,

I know the general form of the Navier Stokes Equation as follows.

I am following a software paper of "Gerris flow solver written by Prof. S.Popinet"
and he mentions after time discretization he ends with the following equation:

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.

Kaat

## Answers and Replies

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The link to the paper doesn't work.

Chestermiller
Mentor
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 thats 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.