up- and downstream regions. To derive the jump conditions we refer the reader to the books of [@Courant1972] and with respect to isothermality to [@Clarke2003]. The RHC rewritten with the relations[^7] \[eq:simvar,eq:densdist\] are given by $$\begin{aligned}
   S_\mathrm{u}\mathcal{U}_\mathrm{u}&=&S_\mathrm{d}\mathcal{U}_\mathrm{d}\nonumber\\
   S_\mathrm{u}\left(\mathcal{U}_\mathrm{u}^2+1\right)&=&S_\mathrm{d}\left(\mathcal{U}_\mathrm{d}^2+1\right),\nonumber
  \end{aligned}$$ where $$\mathcal{U}_\mathrm{u,d}=u_\mathrm{u,d}-\xi_0\label{eq:comov}$$ denote the velocities in the comoving frame with respect to the discontinuity. The velocities $u_\mathrm{u,d}$ are the corresponding velocities in the inertial frame of reference and $\xi_0=\left.\xi\right|_{\left\{r_\mathrm{s},\tau_\mathrm{s}\right\}}$ yields the shock position. These conditions are independent of the radial distance because they vanish in the limit of an infinitely narrow strip and hence are also independent of the number $\alpha$ which distinguishes between the different coordinates. The RHC can be solved for the upstream states $$\mathcal{U}_\mathrm{u}=\mathcal{U}_\mathrm{d}^{-1},~~S_\mathrm{u}=S_\mathrm{d}\mathcal{U}_\mathrm{d}^2,\nonumber$$ if $S_\mathrm{u}\neq S_\mathrm{d}$ and $S_\mathrm{u}, S_\mathrm{d}>0$. These conditions are symmetric, hence exchanging the indices does not change the equations (see \[secBW\]). With the help of one obtains the RHCs with respect to the inertial frame (see also \[fig:Figtube\])

$$\begin{aligned}
    u_{\mathrm{u}}&=\frac{1+u_{\mathrm{d}}\xi_0-\xi_0^2}{u_{\mathrm{d}}-\xi_0}\label{eq:RHCV}\\
    S_{\mathrm{u}}&=S_{\mathrm{d}}\left(u_{\mathrm{d}}-\xi_0\right)^2.\label{eq:RHCS}
  \end{aligned}$$

We emphasize that the numbers $u_{\mathrm{u,d}}, S_{\mathrm{u,d}}$ must satisfy the solutions of . To compute values for the RHC we use `Octave`[^8] with a 4^th^ order Adams-Moulton predictor-corrector scheme [@Hairer1993] and start solving the ODE \[eq:odev\] at the inner boundary \[eq:solv\_ibnd\] up 