\,\, = \,\, {{1}\over{\Delta x}} \,  \int_{\displaystyle x_{j}}
^{\displaystyle x_{j\!+\!1}}
\,W_0(x) \, {\rm d} x \,\,,\qquad  j=0, \cdots,\, J \!-\! 1 \,, \,$

we are able to increment the time step with relation (2.1.10) only if all the fluxes $\,\, f_j^{n+1/2} \,,$ $\,\,j=0,\cdots ,\, J \,$ have been [*a priori*]{} first determined as a functional of the previous values. In a very general way, we say that the finite volume scheme (2.1.10) is an [**explicit scheme**]{} if each flux $\, f^{n+1/2}_j
\,\,$ is a given function $\, \Psi_j \,$ of the mean values $\,\, \bigl( W^n_{k+1/2}
\bigr)_{k=1,\cdots,\, J\!-\!1} \, \,$ at the preceding time step number $\, n
\,$ :

(2.1.12) $\qquad \displaystyle
f^{n+1/2}_j \,\,= \,\,  \Psi_j \bigl(  \, \{ W^n_{k+1/2},\, k=0,\cdots,\, J\!-\!1 \}
\bigr)\,, \qquad j=0,\, \cdots ,\, J  \!-\! 1 \, . \,$

The function $\, \Psi_j \,$ is called the [**local numerical flux function**]{} at point $\, x_{j} \,$ and, joined with the evolution equation (2.1.10), its choice determines the numerical scheme.

$\bullet \qquad \,\,\, $ A natural hypothesis claims that we have [**translation invariance**]{} for the evaluation of the flux if we move the discrete data in the same way ; in other words, the numerical 