fluid problems $\hfill $ 49

$ \qquad 4.5 \quad $ Explicit Runge-Kutta integration with respect to time $\hfill $ 54

[**5) $ \,\, $ References** ]{} $\hfill $ 55

$\bullet \qquad \,\,\, $ [*Acknowledgments*]{}. The author thanks Alexandre Gault, listener at the spring 2000 “lectures in computational acoustics” at the Conservatoire National des Arts et Métiers (Paris, France), for providing his personal manuscript notes.

[1) $ \quad $ Advection equation and method of characteristics. ]{} [1.1 ]{} $ \,  $ [ **Advection equation.** ]{}

$\bullet \qquad \,\,\,\,\, $ We consider a given real number $\,\, a > 0 \,\, $ and we wish to solve the so-called advection equation of unknown function $\,\, u(x,\,t) \,\,$ :

(1.1.1) $\qquad \displaystyle 
{{\partial u}\over{\partial t}} \,\,+\,\, a \,\, {{\partial u }\over{\partial
x}}\,\,=\,\, 0 \,\,, \qquad t \geq 0 \,,\quad x \in \R \,. \,$

We first look to the homogeneity coherence of the different terms of equation (1.1.1). On one hand, the ratio $\,\, {{\partial u}\over{\partial t}} \,\,$ is homogeneous to the dimension $\, [u] \,$ of function $\,\, u({\scriptstyle \bullet},\,
{\scriptstyle \bullet}) \,\,$ divided by the dimension $\, [t] \,$ of the time and we have : $\,\,  {{\partial u}\over{\partial t}} \sim 