introduction to the finite volume method in the context of gas dynamics conservation laws. Our approach is founded on the advection equation, the exact integration of the associated Cauchy problem, and the so-called upwind scheme in one space dimension. It is then extended in three directions : hyperbolic linear systems and particularily the system of acoustics, gas dynamics with the help of the Roe matrix and two space dimensions by following the approach proposed by Van Leer. A special emphasis on boundary conditions is proposed all along the text.

[**AMS Subject Classification:** ]{} 35L40, 35L60, 35L65, 35Q35, 76N15.

[**Keywords:**]{} Advection, Characteristics, Roe matrix, Van Leer method.

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[Contents ]{}

[**1) $ \,\, $ Advection equation and method of characteristics** ]{}

$ \qquad 1.1 \quad $ Advection equation $\hfill $ 3

$ \qquad 1.2 \quad $ Initial-boundary value problems for the advection equation $\hfill $ 4

$ \qquad 1.3 \quad $ Inflow and outflow for the advection equation $\hfill $ 6

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

$ \qquad 2.1 \quad $ Linear advection $\hfill $ 8

$ \qquad 2.2 \quad $ Numerical flux boundary conditions $\hfill $ 13

$ \qquad 2.3 \quad $ 