The Sod's Problem
The first example considered here is known as the Sod's Problem. The initial data are:
Being the left pressure greater than the right one, the pressure discontinuity results in a shock wave which will propagate rightward. In addition, the central contact discontinuity, visible in the density plot, propagates rightward, while a rarefaction wave propagates leftward from the origin.
The solution of this problem has a monotone decreasing density profile and therefore it does not display certain difficulties that may arise when the intermediate state has to be ``built-up'', see Harten et. al (1987).
The Lax's Problem
To bypass this problem, a second test case is considered where the initial data are:
This last test case is known as Lax's Problem, Lax (1954). ,
You can also try to build your own test case, by changing the input data. The program will always show the exact solution for all the variables involved.