dashed dotted lines are singular lines. Their intersection gives a saddle (triangle) and an unstable improper node (dots). Thick lines starting at singular points are selected solutions. []{data-label="fig:pp_a1k0"}](Figure02.pdf)

### Cartesian coordinates and 

This constitutes the simplest case because the ODEs \[eq:odev,eq:odes\] reduce to constant solutions separated only by the discontinuity. Hence, we are left with calculating the RHCs \[eq:RHCV,eq:RHCS\] using the method described in \[secRHC\]. All the numbers are collected in \[tab:numerRHC\]. The solutions, yielding a simple step function are plotted in figure \[fig:solk0comb\]a,b. The graphs show them back transformed with the similarity variable $\xi\mapsto r=\xi\cdot\tau$ (), so the shock position refers to $r=r_\mathrm{s}$ at different times $\tau$ again. For comparison, we mention the solutions for non-isothermal cartesian shocktubes [@Meyer1988; @Gehmeyr1997].

### Polar and spherical coordinates and  {#polshperea1k0}

The discussion for polar and spherical coordinates and a constant initial density distribution requires more effort. We plot the phase diagrams \[fig:pp\_a1k0,fig:pp\_a2k0\] together with the singular lines $u\left(\xi\right)=\left\{0\wedge\xi\pm 1\right\}$ and the singular points at $\mathcal{D}_1=\left(0,0\right)$ for both coordinates. These singular points are both saddle points with their stable solutions receding along the eigenvector $\boldsymbol{x}_2=\left(1,0\right)^\mathrm{T}$ and their unstable solutions leaving along $\boldsymbol{x}_1=\left(0,1\right)^\mathrm{T}$. The points $\mathcal{D}_2=\left(1,0\right)$ show different behavior. For $\alpha=1$ we find an 