\R \,, \,\,$

and we search a discrete version of this mathematical model. For doing this, we introduce a space step $\, \Delta x > 0 \,$ and a space grid composed by points $\,\,  x_{j} \,\,$ whose coordinates are multiples of this space step $\,
\Delta x ,\,$ [*id est* ]{}

(2.1.2) $\qquad \displaystyle
 x_{j} \,=\, j \, \Delta x \,, \qquad j \in \Z \,.  \,$

For a finite domain, $\, ]0,\,L[ \,$ to fix the ideas, the above grid is limited to integer values $\,j \,$ such that

(2.1.3) $\qquad \displaystyle
0 \, \leq \, j \, \leq \, J \,=\, {{L}\over{\Delta x}} \,$

and the vertices $\,\, (x_j)_{0\leq j \leq J} \,$ are usually used in the context of the finite difference method. The intervals $\,\, K_{j+1/2} \,=\,\,
 ]x_j,\,x_{j+1} [\,\,$ between two vertices can be considered as finite elements (or finite volumes in our study) and they cover the entire domain $\, ]0,\,L[  \,: \,$

(2.1.4) $\qquad \displaystyle
[ 0,\,L ] \,\,= \,\, \bigcup_{ 0 \leq j \leq J \! - \! 1} [x_{j} \,,\, x_{j+1} ] 
\, \,,\, $

as proposed in the general context of meshes (see [*e.g.*]{} Ciarlet \[Ci78\]). We introduce also a 