\!-\! 1/2 \,.$ If $\, a < 0 ,\,$ the propagation of the information with the advection equation is from right to left ; the interface flux at the abscissa $\,
x_{j} \,$ is due to the control volume on the right, [*i.e.*]{} with number $\, j\!+\!1/2
\,$ as depicted on Figure 2.2.

=3,5cm

[**Figure 2.2.**]{} Upwinding of the information for the advection equation.

=2,5cm

[**Figure 2.3.**]{} Notations for the one-dimensional finite volume method.

$\bullet \qquad \,\,\, $ Recall that practical use of the upwind finite volume scheme like (2.1.17) when $\, a >
0 \,$ or (2.1.18) if $\, a < 0 \,$ is restricted to the usual Courant-Friedrichs-Lewy stability condition :

(2.1.21) $\qquad \displaystyle
a \, {{\Delta t}\over{\Delta x}} \,\, \leq \,\, 1 \,$

as developed [*e.g.*]{} in the book of Richtmyer and Morton \[RM67\].

[2.2 ]{} $ \, $ [ **Numerical flux boundary conditions** ]{}

$\bullet \qquad \,\,\,\,\, $ In this section, we focus on the problem of the numerical boundary conditions. Recall that we study the advection equation in the space domain $\,\, [0,\,L] \,: \,$

(2.2.1) $\qquad \displaystyle
0 \, \leq \, x \, \leq \, L \, $

and $\,\, J \,= \, {{L}\over{\Delta 