The Riemann problem in linearized gas dynamics
As a simple example, we investigate the properties of the one-dimensional Riemann problem
in gas dynamics, with initial conditions 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:
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 :
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 in a computer simulation. Generally as more conservation equations are used, more characteristics are involved.