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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"

[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:

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 u

_{n+0.5}and

u

_{n+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.