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

=3,7cm

[**Figure 1.5.**]{} Initial-boundary value problem for the advection equation

with $\,\, a > 0 \,\,$ in the domain $\,\, 0 < x < L \,\,$ and $\,\, t > 0. \,\,$

=4,5cm

[**Figure 1.6.**]{} Initial-boundary value problem for the advection equation

with $\,\, a < 0 \,\,$ in the domain $\,\, 0 < x < L \,\,$ and$\,\,
t > 0. \,\,$

$\bullet \qquad \,\,\, $ The difference between point $\, x=0 \,$ and point $\, x=L \,$ for the resolution of the advection equation in space-time domain (1.3.1) is due to the fact that we choose an orientation of the characteristic lines $\,\, x-at \,=\, {\rm constant} \,\,$ associated to an [ **increase**]{} for the time direction. With this choice of time direction, the characteristic lines [**enter**]{} inside the space-time domain (1.3.1) at $\,\, x=0 \,\,$ and they go outside at $\,\, x \!=\! L \,. \, $ The boundary condition (1.3.3) is given at the input of the domain (see Figure 1.5) and at $\,\,x
\!=\! L \,,\,$ there is a free output from space time domain (1.3.1), without necessity to specify any numerical boundary condition.

$\bullet \qquad \,\,\, $ If we 