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The shock-tube problem is a very interesting test case because the exact time-dependent
solution is known and can be compared with the solution computed applying
numerical discretizations. The initial solution of the shock-tube problem is composed by
two uniform states separated by a discontinuity which is usually located at the origin.
This particular initial value problem is known as Riemann Problem. The initial left and
right uniform states are usually introduced by giving the density, the pressure and
the velocity. This initial set represents a tube where the left and the
right regions are separated by a diaphragm, and filled by the same gas in two
different physical states. If all the viscous effects are negligible along the tube walls
and assuming that the tube is infinitely long in order to avoid reflections at the tube ends,
the exact solution of the full Euler equations can be obtained on the basis of a simple
wave analysis. At the bursting of the diaphragm, the discontinuity between the two initial
states breaks into leftward and rightward moving waves, wich are separated by a contact surface.
Each wave pattern is composed by a contact discontinuity (C) in the middle, and a shock (S)
or a rarefaction wave (R) at the left and the right sides separating uniform state
solution. All the available combinations produce four wave patterns: RCR, RCS, SCR,SCS,
which are self-similar, that is it depends only on x/t. A fifth pattern is possible in theory,
and it contains a vacuum state between two central contact discontinuities, which occur
between two rarefaction waves. This case is of only theoretical interest, because it is
the limit of the perfect gas equations at zero pressure and temperatures, but it can never
occur in practice.