CFD Online Discussion Forums - Hyperbolic equasion solver or How OF works

Hello,
I have working hyperbolic equasion solver (hyper1Foam in attachment - works in OF 2.1.0).

Main equation:
$\frac{\partial\psi }{\partial t}+C{\frac{\partial \psi}{\partial x}}=0$
accordingly

Code:

        solve

        (

            fvm::ddt(psi)

            +

            fvm::div(phi,psi)

        );

and in fvShemes

Code:

ddtSchemes

{

    ddt(psi) Euler;

}

divSchemes

{

    div(phi,psi) Gauss linear;

}

Working geometry - coube 1x1x1 diveded on 5 equal parts along oZ axe.

Main equation in discretized form:

[LaTeX Error: Syntax error]
$C_i=\frac{V\phi_i}{\Delta t}$
where
$\psi_i - old ~value ~of ~phi ~in ~cell ~i$
$\psi^n_i - new ~value$

In my case

V = const = 1/5 = 0.2 ~~ volume ~of ~single ~cell

$\Delta t = 0.1 ~time ~step$

And when I insert result of phi from one of time directories...

$for ~\phi_i = 0,434805$
$\frac{0.2 *0,749653 - 0.2*0,434805}{0.1 } + (0,2/0,1)*0,434805*{\frac{0,206803 - 1}{0.2}} =$

Not zero!
I khow that hyperbolic equation will not =0, because it is iterational method. Is it residual?

I want to understand - how OpenFOAM count values of phi from one iteration to another - how to get new phi from old values:
phi_new = ... ... ...

And what initial values of phi?

Results: Table of results on GoogleDocs