x}} \, \in \N \,\,$ control cells (or finite elements) have been used to define a mesh :

(2.2.2) $\qquad \displaystyle
J \, \Delta x \,\, = \,\, L \,.\,  $

Note that the $\,j^{\rm th} \,$ cell is exactly the interval $\,\, ]x_{j-1} ,\,
x_{j}[ \,\,$ and it is centered at point $\,\, x_{j-1/2}\,$ as shown on Figure 2.3.

$\bullet \qquad \,\,\,\,\, $ At time step $\, n \, \Delta t \,,\,$ the discrete field is entirely known and is composed of all the values $\,\, W^n_{j-1/2} \,$ for $\,\, j\!=\! 1,\, \cdots ,\, J \,.
\, $ With a flux function $\,\, \Phi({\scriptstyle \bullet} ,\, {\scriptstyle \bullet}) 
\,\,$ as proposed at relation (2.1.14), we observe that the two boundary fluxes $\,\,
f^{n+1/2}_0 \,$ and $\, f^{n+1/2}_J \,$ are [**not**]{} a priori defined because states $\, W^n_{-1/2} \,$ or $\, W^n_{J+1/2} \,$ does not exist. The situation is more complex with numerical fluxes that use four points or more as proposed in (2.1.13) and will not be detailed in this section. Even if the formula giving the numerical flux at the boundaries has to be specifically studied, the finite volume scheme remains defined by the relation (2.1.10) and we 