$2.19 \cdot 10^ {-5}$    $4.18 \cdot 10^ {-6}$     $2.38 \cdot 10^ {-5}$     $3.68 \cdot 10^ {-6}$     $3.78 \cdot 10^ {-5}$
          1/3         0       $1.55 \cdot 10^ {-5}$    $1.37 \cdot 10^ {-5}$     $2.27 \cdot 10^ {-5}$     $5.95 \cdot 10^ {-6}$     $1.03 \cdot 10^ {-5}$
   EP     1/2                 $1.62 \cdot 10^ {-5}$    $3.09 \cdot 10^ {-6}$     $2.45 \cdot 10^ {-5}$     $3.56 \cdot 10^ {-6}$     $1.36 \cdot 10^ {-6}$
          2/3                 $1.92 \cdot 10^ {-5}$    $7.16 \cdot 10^ {-6}$     $2.67 \cdot 10^ {-5}$     $5.92 \cdot 10^ {-7}$     $1.24 \cdot 10^ {-5}$

  : Comparison between the numerical (\[secRHC\]) and analytical (\[anasolIP,anasolEP\]) computation of the RHC. We state very good agreement and hence correctness of the numerical method. []{data-label="tab:vergleichRHC"}

In general, the singular lines divide the phase space in disjoint regions. None of the solutions from one region connects continuously to a solution in the other region [@Sedov1959]. Thus a solution stretching across the whole spatial domain must contain a discontinuity. Possible solutions shown in the phase diagrams \[fig:pp\_a1k0,fig:pp\_a2k0,fig:pp\_a1k1,fig:pp\_a2k1,fig:pp\_blast\_k0,fig:pp\_kink\] are no continuous solutions, they simply end at some joint position at the singular line.

![Phase diagram for polar coordinates and constant density distribution ($\alpha=1, \kappa=0$). The thin solid curve denotes the separation of the domain by $\xi=0$. The 