question is how to choose a numerical finite volume scheme. The simplest choice consists in a two point explicit scheme such that the finite difference scheme is identical to the upstream-centered scheme (see [*e.g.*]{} Richtmyer-Morton \[RM67\]). It takes the following expressions :

(2.1.17) $\qquad \displaystyle
{{1}\over{\Delta t}} \bigl( W_{j+1/2}^{n+1} - 
W_{j+1/2}^{n} \bigr) \,\,+ \,\, a \Bigl( W_{j+1/2}^{n} - W_{j-1/2}^{n} \Bigr) \,\, =
\,\, 0 \,, \qquad a > 0 \,$

(2.1.18) $\qquad \displaystyle
{{1}\over{\Delta t}} \bigl( W_{j+1/2}^{n+1} - 
W_{j+1/2}^{n} \bigr) \,\,+ \,\, a \Bigl( W_{j+3/2}^{n} - W_{j+1/2}^{n} \Bigr) \,\, =
\,\, 0 \,, \qquad a < 0\,. \,$

The corresponding flux function is called the [**first order upstream-centered flux**]{}, is simply given by the following relations :

21= 22= 30= 31= 44=

$ \box44 $

When this flux function acts at a given point $\, x_{j} \,$ of the mesh, we have :

21= 22= 30= 31= 44=

$ \box44 $

If $\, a>0 , \,$ the exact solution of the advection equation propagates the information from the left to the right ; the flux at the interface $\, x_{j} \,$ is issued from the cell at the left of the interface and this cell at the number $\,
j 