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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?
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 $\nabla \cdot U^{n+1} = 0$.
