# How to deal with time derivative of speed toward time in staggered grid in CFD?

In my control equation, there is a term $$\frac{\partial \rho u}{\partial t}+\frac{\partial \rho u^2}{\partial x}+\cdots=\frac{\partial p}{\partial x}$$

How to discretize it in staggered grid?

For convenience, I include a picture of a staggered grid: If u is replaced by a scalar variable like $$p$$, we can simply integrate it in the scalar CV and write $$p_P-p_P^o$$ in which superscript $$o$$ means old time value. However in staggered grid, we store speed at cell face $$w,e,n,s$$ of a scalar CV then how can we write it in discrete form?

Basically I don't know how should we integrate this equation because there are two types of variables involved.

I can't believe that I forgot in SIMPLE we first treat $$p$$ as known.
The we should integrate this equation over the u-control volume. Because the time term no relation to $$x$$ and $$y$$, so it's simply multiplied by $$\Delta V$$. Thus we can write $$\rho (u_w-u_w^o)\Delta V$$