First we have Discretization of the Transport Equation
$$
\frac{\partial \rho \phi}{\partial t} + \nabla(\rho U \phi) - \nabla (\rho \Gamma\_\phi \nabla \phi) = S\_\phi (\phi)
$$

In Finite Volume Method it looks like:
$$
\int\_t^{t+\Delta t} \left[ \frac{\partial}{\partial t} \int\_{V\_p} \rho \phi dV + \int\_{V\_p} \nabla \cdot (\rho U\phi)dV - \int\_{V\_p} \nabla \cdot (\rho \Gamma\_\phi \nabla \phi)dV \right]dt = \int\_t^{t+\Delta t} \left(\int\_{V\_p}S\_\phi(\phi)dV \right)dt
$$

and after transformations:

$$
\int\_t^{t+\Delta t} \left[ \left( \frac{\partial \rho \phi}{\partial t} \right)\_p V\_p + \sum\_f F\phi\_f - \sum\_f(\rho \Gamma\_\phi)\_f S.(\nabla\phi)\_f \right]dt=\int\_t^{t+\Delta t}(SuV\_p+S\_pV\_p\phi\_p)dt
$$

and time discretization:
$$
\frac{\rho\_p\phi\_p^n-\rho\_p\phi\_p^o}{\Delta t}V\_p+ \frac{1}{2}\sum\_f F\phi\_f^n-\frac{1}{2}\sum\_f(\rho\Gamma\_\phi)\_f S.(\nabla\phi)^n\_f+ \frac{1}{2}\sum\_f F\phi\_f^o-\frac{1}{2}\sum\_f(\rho\Gamma\_\phi)\_f S.(\nabla\phi)^o\_f = SuV\_p + \frac{1}{2}S\_pV\_p\phi\_p^n+\frac{1}{2}S\_pV\_p\phi\_p^o
$$

For every cell we can make equation:

$$
a\_p\phi^n\_p+\sum\_N a\_N \phi\_N^n = R\_p
$$

But how we can get elements of matrix $A$ if we don't know $\phi$?
$$
[A][\phi]=[R]
$$

Here Hrvoje Jasak wrote that every coefficient $ a\_p $ includes the contribution from temporal derivative, convection and diffusion terms. But what formula of $a\_p = ...$?

<http://powerlab.fsb.hr/ped/kturbo/OpenFOAM/docs/HrvojeJasakPhD.pdf>