IP and different numbers for $\kappa$. []{data-label="tab:numerRHC"}

We derive analytical solutions to both, the IP and the EP in slab geometry in . The numerical values from \[tab:numerRHC\] are checked against the analytical values derived by the corresponding methods with $$\delta X=\frac{X_\mathrm{num}-X_\mathrm{analyt}}{X_\mathrm{analyt}}$$ for every $X\in\left\{\xi_0,u_\mathrm{u},u_\mathrm{d},S_\mathrm{u},S_\mathrm{d}\right\}$. The results are summarized in showing a deviation of about $\delta X\approx 10^{-5}\mbox{\ to\ }10^{-6}$ which nicely demonstrates the accuracy of the described numerical method. The relative error of the numerical integration method is set to $10^{-6}$, such that the error bound is well suited. The error bound is primarily justified by the precision of $\delta X$ but also by the computing time of the algorithm.

Self-similar solutions {#sec:ssimsol}
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We discuss the general method for solving numerically. We give the solutions for $\kappa=0$ and $\kappa=1$ as examples in the different coordinates ($\alpha\in\left\{0,1,2\right\}$) and discuss further restrictions to $\kappa$. Solutions for the genuinely isothermal explosion problem in spherical coordinates are considered afterwards.

Numerical solutions in arbitrary coordinates {#kappa=0}
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We achieve numerical solutions for all possible cases by numerical integration of the ODEs \[eq:odev,eq:odes\]. We take advantage of the library `ode`[^10] [@Tufillaro1992], which enables us to compute solutions with well known numerical solvers and with 