Verification of analytical derivation. Each plot shows the IC, as well as the simulation run using solve_ivp in Python, zeroth-order solution T⁽⁰⁾, and first-order solution T⁽¹⁾, all at a later time such that χt >> 1. The colored ‘+’ corresponds to the HWHM point of each calculation. This is done for increasing background values T_∞/T₀ given by the plot titles. The simulation box extends to r = 100 R₀ in (c), preventing any impacts caused by the boundary condition.