( a
\, W \bigr) (x_{j},\,t) \Bigr] \, {\rm d}t \,\,= \,\, \Delta t \,\, \Bigl[ \,
f_{j+1}^{n+1/2} - f_{j}^{n+1/2} \, \Bigr] \, \,$

according to the definition (2.1.9). We add the two previous results, use identity (2.1.7) and divide by $\,\, \Delta t \, \Delta x .\,\,$ We obtain exactly the relation (2.1.10). $ \hfill \square \kern0.1mm $

$\bullet \qquad \,\,\, $ The relation (2.1.10) is a very general form for the evolution of the mean values $\,\, W_{j+1/2} \,\,$ between two time steps. The increment $\,\,  \bigl(
W_{j+1/2}^{n+1} -  W_{j+1/2}^{n} \bigr) \,\, $ is, after correction by a multiplicative factor, equilibrated by the flux difference $\,\, \bigl( f_{j+1}^{n+1/2} - f_{j}^{n+1/2} \bigr) \,.\,$ The idea of a finite volume scheme is to consider now that the algebraic object $\,\,  W_{j+1/2} \,\,$ is nomore the mean value of the exact solution but an [**approximation**]{} of this mean value. Then the relation (2.1.10) proposes a numerical scheme for the discrete evolution of the approximated mean values $\,\,  W_{j+1/2} \,,\,\, j=0, \cdots,\,
J \!-\! 1 .\,$ Nevertheless, the numerical scheme is [**not**]{} entirely defined by the relation (2.1.10). Starting from mean values at the initial time step, [*i.e.*]{}

(2.1.11) $\qquad \displaystyle
W^0_{j+1/2} 