text: As a simple example, we investigate the properties of the one-dimensional Riemann problem 
  in gas dynamics 
  (Toro, Eleuterio F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics, Pg 44, Example 2.5)
  The initial conditions are given by
  where x = 0 separates two different states, together with the linearised gas dynamic equations (see gas dynamics for derivation).
  where we can assume without loss of generality .
  We can now rewrite the above equations in a conservative form:
  where
  and the index denotes the partial derivative with respect to the corresponding variable (i.e. x or t).
  The eigenvalues of the system are the characteristics of the system
  . They give the propagation speed of the medium, including that of any discontinuity, which is the speed of sound here. The corresponding eigenvectors are
  By decomposing the left state  in terms of the eigenvectors, we get for some 
  Now we can solve for  and :
  Analogously
  for
  Using this, in the domain in between the two characteristics ,
  we get the final constant solution:
  and the (piecewise constant) solution in the entire domain :
  Although this is a simple example, it still shows the basic properties. Most notably, the characteristics decompose the solution into three domains. The propagation speed
  of these two equations is equivalent to the propagation speed of sound.
  The fastest characteristic defines the Courant–Friedrichs–Lewy (CFL) condition, which sets the restriction for the maximum time step for which an explicit numerical method is stable. Generally as more conservation equations are used, more characteristics are involved.