\,\, t > 0 \,,\,$

is advected inside the domain of study and we have :

(1.3.7) $\qquad  \displaystyle 
u(x, \,t) \,\,\, = \,\, w_{L} \Bigl( t \,+\, {{L}\over{a}}  \,-\, {{x}\over{a}}\Bigr)
\,\,, \qquad x-at \,>\,L  \,. \,$

We have established the following

[**Proposition 1.3. $\quad$ Advection in the domain $\, 0<x<L \,,\,\, a < 0 .\,$** ]{}

Under the hypothesis (1.3.4), the resolution of the advection equation (1.2.1) in the space-time domain (1.3.1) conducts to a [**well posed**]{} problem when we introduce the initial condition (1.3.2) on the interval $\, ]0,\,L [ \,$ and the boundary condition (1.3.6) at the [**input**]{} region located at $\, x \!=\! L  \,,\, $ without any boundary condition at the output located at $\, x =0 \,.\,$ The solution of Problem (1.2.1), (1.3.2) and (1.3.6) is given by the relations (1.3.5) and (1.3.7).

[2) $ \,\,\,\,\,\,\, $ Finite volumes for linear hyperbolic systems.]{} [2.1 ]{} $ \,  $ [ **Linear advection.** ]{}

$\bullet \qquad \,\,\,\,\, $ We still study the advection equation parameterized by some celerity $\, a > 0 \,$ :

(2.1.1) $\qquad \displaystyle
{{\partial W}\over{\partial t}} \,\,+\,\, {{\partial }\over{\partial x}} \bigl(
a \, W \bigr)  \,\,=\,\, 0 \,\,,\qquad t>0 \,,\qquad x \in 