Thermal equilibrium (ṁ–Σ) diagram for slim-disc solutions at R = 30 M. Red dots: Numerical solutions from the slim-disc code (Sa¸dowski 2011). Black line: analytical S-curve, obtained from the Paczyński-Wiita potential. Both sets have M = 10 M_⊙ and α = 0.01. The analytical S-curve has also ξ = 1, Ω/Ω_K = 1. Green crosses: obtained from the Paczyński-Wiita potential but with the same ξ, Ω/Ω_K values as the corresponding slim-disc simulations. Although the black line does deviate from the red points, we see that when the above corrections are taken into account, the analytical pseudo-Newtonian slim discs agree with the stationary, general relativistic, “1+1” simulations with great precision.