flux function $\,\, \Psi_j \, \,$ only depends on the $\,p
\,$ first neighbors of the interface $\, x_{j} \,.\,$ Then the explicit numerical flux is a given function $\, \Phi \,$ of the $\, p\, $ first neighbors and we have :

(2.1.13) $\qquad \displaystyle
f^{n+1/2}_j \,\,= \,\,  \Phi \bigl( W^n_{j+1/2-p}  ,\,\cdots ,\, 
W^n_{j-1/2} ,\, W^n_{j+1/2} ,\, \cdots ,\,  W^n_{j+1/2+p-1} \bigr) \,. \,$

A very important particular case is one of a two-point scheme for the evaluation of the numerical flux. We have in this particular case :

(2.1.14) $\qquad \displaystyle
f^{n+1/2}_j \,\,= \,\,  \Phi \bigl( W^n_{j-1/2} ,\, W^n_{j+1/2}  \bigr) \,. \,$

With this particular choice, the numerical scheme for incrementing in time of the mean values takes the form :

21= 22= 30= 31= 44=

$ \box44 $

It is also a three-point finite difference scheme. The finite volume scheme (2.1.10) (2.1.13) is said to be [**consistent**]{} with the advection equation (2.1.1) when the numerical flux function $\, \Phi \,$ satisfies the condition

(2.1.16) $\qquad \displaystyle
\Phi \bigl( W  ,\,\cdots ,\,  W ,\, W ,\, \cdots ,\,  W \bigr) \,\, = \,\, a \, W
\,\,,\qquad \forall \, W \in \R \,.$

$\bullet \qquad \,\,\, $ The crucial 