Riemann Problem (Gas dynamics)

Riemann and Godunov problems (at the first step) with initial conditions piecewise constant

Numerical solution of the Riemann problem with initial conditions piecewise constant. Problem shock tube in which at the time t=0- two states have been defined: u1, a1, p1 and u4, a4, p4. At the time t=0+ a septum separating the two regions is raised impulsively.We determine the states 2 and 3, debating between the 4 possible types solution : RCR , RCS , SCR, SCS .
In particular, if the initial data are such as to induce a type of solution of the type NCR , NCS, RCN , SCN, then the program provides such a type and it is possible to deduct from the results, whereby at example the speed of sound of three contiguous states is equal , as long as the input of initial data having an accuracy up to the digit decimal significant error.
Example: state 1: u1=0.00000000000 a1=1.00000000000 p1= 1.00000000000
state 4: u4=1.94747747000 a4=1.38949549000 p4=10.00000000000
The solution is NCR , but the program could provide SCR if the datas are not precise enough to a suitable digit , however, is known as the "shock" on u-a really is nothing in intensity , it is as if there wasn't , in fact you get a Mach its upstream ( and downstream ) of to 1 . And yet you will have a1 = a2 = a3 . So be careful and to correlate the type of solution with the correct interpretation of the results !
WE STRONGLY RECOMMEND TO ENTER DATA WITH AN INITIAL ACCURACY UP TO THE AMOUNT REPORTED SIGNIFICANT DECIMAL ERROR.
GODUNOV's METHOD
The following part of hte program provides an useful tool to examine a Riemann's sub-problem, or the evolution of a discontinuity at from piecewise constant initial conditions. It provides a graph on the plane (x,t) of the system of waves that are generated. It is also performed the calculation of the ratio tau/h = alpha/lambda_max necessary to satisfy the condition for which at the interface, the flow of characteristic variables of the method of godunov is constant for the time analysis.
The domain of interest is divided into an appropriate number of cells, in particular, we intend to analyze the temporal evolution of variables at the first time step, and then, having one discontinuity and having imposed CFL will be only two cells affected from the system of waves (those adjacent the discontinuity). It will put the discontinuity on the interface (j-1/2) and then to the first step of integration is not trivial only the analysis of the cells (j-1) and (j). The cells prior to the (j-1) and subsequent to (j) will have a state coincident with the initial one (respectively 1 and 4).