time step $\,\Delta t >0 \,$ and the discrete time values at integer multiples of the above quantum :

(2.1.5) $\qquad \displaystyle
t^n \,=\, n \, \Delta t \,, \qquad n \in \N \,.  \,$

We consider now a space-time volume $\,\,V_{j+1/2}^{n+1/2} \,\,$ obtained by cartesian product of the two intervals $\,\, ]x_{j} \,,\, x_{j+1}[ \,\,$ and $\,\,
]t^n ,\,t^{n+1} [\,\,$ (see Figure 2.1) :

(2.1.6) $\qquad \displaystyle
V_{j+1/2}^{n+1/2} \,\,= \,\,  ]x_{j} \,,\, x_{j+1}[  \, \times \, ]t^n ,\,t^{n+1}
[ \,. \,\,$

$\bullet \qquad \,\,\,\,\, $ The finite volume scheme consists simply in integrating the advection equation (2.1.1) inside the space-time domain $\,\, V_{j+1/2}^{n+1/2} \,\,$ introduced previously :

(2.1.7) $\qquad \displaystyle
\int_{ \displaystyle V_{j+1/2}^{n+1/2}} \, \biggl[ \, 
{{\partial W}\over{\partial t}} \,\,+\,\, {{\partial }\over{\partial x}} \bigl(
a \, W \bigr) \, \biggr] \, {\rm d}x \, {\rm d}t \,\, =\,\, 0 \,\,,\quad 0 \leq j
\leq J \,,\qquad n \geq 0 \,.  \, $

[**Figure 2.1.**]{} Space-time grid for the finite volume method.

[**Proposition 2.1. $\quad$ Finite volume scheme.**]{}

Let $\,\, \R \times [0,\,+\infty[ \, \ni (x,\,t) \longmapsto W(x,\,t) \in \R \,\,$ be a solution of the advection equation (2.1.1). We introduce the space mean value $\,\,W_{j+1/2}^{n} \,\,$ of this solution 