give well defined and finite numbers for the BCs. The velocity equation \[eq:odev\] then becomes $$\frac{\mathrm{d}u}{\mathrm{d}\eta}=\frac{\kappa-u\alpha\eta}{\left(1-u\eta\right)^2-\eta^2}\label{eq:veleta}$$ and the solution of the approximate equation $$\frac{\mathrm{d}u}{\mathrm{d}\eta}\approx \kappa-u\alpha\eta\mbox{,\ \ for\ }\quad 0<\eta\ll1\quad\mbox{and}\quad -1\leq\left|u\right|\leq0$$ is written in terms of the imaginary error function $\mathrm{erfi\left(z\right)}$ $$u\left(\eta\right)=e^{\frac{-\alpha\eta^2}{2}}\left(c_1+\sqrt{\frac{\pi}{2\alpha}}\kappa\mathrm{\ erfi}\left(\sqrt{\frac{\alpha}{2}}\eta\right)\right)$$ with the constant of integration $c_1$. The IP requires a constant negative inflow velocity complying with [@Noh1987], and the EP a vanishing velocity field as $\eta=\xi^{-1}\rightarrow0$ complying with [@Sedov1946]. Hence, the BCs are fixed with $$\begin{aligned}
    u_{\infty}&= \lim_{\eta\to0}u\left(\eta\right)\equiv c_1=\left\{
    \begin{array}{ll}
      -1 & \mbox{for\ IP}  \\
       0 & \mbox{for\ EP}
    \end{array}\right. .\label{eq:ubnd}
  \end{aligned}$$ Applicability to IPs or EPs requires the RHC to be satisfiable under above BCs. Therefore, the outer BCs do not yield any loss of generality. The formerly six auxiliary conditions have been reduced to the two conditions on $\left(u_\infty,S_\infty\right)$ stating a well posed boundary value problem to the set of ODEs \[eq:odev,eq:odes\]. The conditions $\left(u_0,S_0\right)$ emerge from solutions of the ODEs and application of the RHC.

Isothermal Rankine-Hugoniot Conditions {#secRHC}
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As indicated in , the RHC are required to describe the discontinuity created by the shockwave. These conditions are only valid in an infinitesimal area at the discontinuity and connect the adjacent states of the 