to some test value $\tilde\xi_0=\tilde\eta_0^{-1}$ as an initial guess for the jump position. This gives the number $u_\mathrm{d}(\tilde\xi_0)=\tilde u_\mathrm{d}$. Then we integrate Eq. \[eq:veleta\] from the outer BC \[eq:ubnd\] up to the value $u_\mathrm{u}(\tilde\eta_0)=\tilde u_\mathrm{u}$ and application of the RHC \[eq:RHCV\] yields $\tilde u_\mathrm{d}^*$. We are now able to compute the relative error $$\epsilon=\left|\frac{\tilde u_\mathrm{d} - \tilde u_\mathrm{d}^*}{\tilde u_\mathrm{d}^*}\right|\leq10^{-4}\label{eq:errbnd}$$ with $\epsilon$ as a sufficient error bound. By checking over- or under estimation of the value $\tilde\xi_0$ we increase or decrease the next guess by $10\%$, staying close to the relevant region[^9] (see e.g. \[polshperea1k0\]). The procedure then is repeated until a sufficient $\epsilon$ is found and hence converges to the exact RHC. The numbers for some particular solutions computed for different geometries are given in \[tab:numerRHC\].

 

   $\kappa$   $\alpha$   $\xi_0$   $u_{\mathrm{u}}$   $u_{\mathrm{d}}$   $S_{\mathrm{u}}$   $S_{\mathrm{d}}$
  ---------- ---------- --------- ------------------ ------------------ ------------------ ------------------
      0          0       0.6181          -1.0               0.0                1.0              2.61780
                 1       0.7371        -0.6196              0.0               2.6802             4.9328
                 2       0.8376        -0.3562              0.0               5.4619             7.7844
     1/3         0       0.9388        -0.7906             0.3606             0.8507             2.5447
     1/2                 1.1905        -0.7262             0.6688             0.8175             3.0030
     2/3                 1.5829        -0.7017             1.1452             0.8004             4.1777
      1          1       1.3657        -0.3028             0.7663             1.2373             3.4443
                 2       1.2594        -0.0445             0.4925             2.1326             3.6260

  : Numerical values for the RHC corresponding to the 