My professor gave me the following discretization scheme to use to discretize the mass conservation equation $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$. I haven't seen this scheme used anywhere and was wondering how this was derived. And if I use the more popular Taylor series expansion for partial differential equations, will it work?
$(\frac{\partial u}{\partial x})_{i,j} = \frac{1}{2}(\frac{u_{i,j}-u_{i-1,j}}{\Delta x}+\frac{u_{i,j-1}-u_{i-1,j-1}}{\Delta x})$