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I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf

It says applying the divergence to both sides of this equation $$\frac{1}{\Delta t} U^{n+1} - \frac{1}{\Delta t} U^{n} = - \nabla P^{n+1} $$ yields the linear system, $$-\Delta P^{n+1} = - \frac{1}{\Delta t} U^{n} $$

Why does the $\frac{1}{\Delta t} U^{n+1}$ term vanish by taking the divergence?

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  • $\begingroup$ Since you're solving the incompressible NS equations, the real velocity fields (as opposed to the intermediate velocity fields before pressure-correction) are divergence-free by definition. The wiki page for this solution method is pretty informative here. en.wikipedia.org/wiki/… $\endgroup$ – Tyler Olsen Oct 19 '16 at 11:06
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It is a consequence of the continuity condition ($\nabla \cdot u = 0$), condition (3) at page 2.

At chapter 4 at the beginning there is the assumption about the condition (3) is valid. You are at the $\textit{n+1}$, step and so $$\nabla \cdot U^{n+1} = 0$$

This method is in the family of projection method. In this class of methods the pressure and velocity problems are decouple. The intermediate velocity does not respect the continuity condition. The method use the pressure to impose the continuity condition, i.e. the pressure is used to project the intermediate velocity onto a space of divergence-free velocity field. So when you calculate the pressure use $\nabla \cdot U^{n+1} = 0$.

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  • $\begingroup$ Shouldn't the continuity condition be satisfied at the every time step? So it also should be $\nabla U^{n}=0$. But in this case $\nabla U^{n}=0$ is the intermediate velocity, so it does not satisfy the condition. Is my understanding correct? $\endgroup$ – user26767 Oct 19 '16 at 11:37
  • $\begingroup$ I edit the answer, but You have already understood the focal point. $\endgroup$ – Mauro Vanzetto Oct 19 '16 at 13:41

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