An introduction to finite volumes for gas dynamics

We propose an elementary 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.