Non-LTE transfer - IV. A rapidly convergent iterative method for the Wiener-Hopf integral equations

A simplified solution of the Wiener–Hopf equation of non-coherent transfer introduced in Frisch & Frisch (1975) is extended into a systematic approximation procedure by an iterative under-relaxation method. Contrary to the Λ-iteration, the number of iterations required to achieve a given accuracy is independent of ∊. The order of interpolation of the source function between mesh-points needed in the numerical calculations depends on the scattering kernel: for a Doppler profile at least quadratic interpolation is required. It is shown that the Finn & Jefferies (1968) step-wise constant representation of the source function is in fact equivalent to a quadratic interpolation. After suitable modifications the method is also applied to the singular integral equation obtained for the interior solution by Frisch & Frisch (1977) when $$\epsilon \rightarrow 0$$ and to Ivanov's homogeneous Wiener–Hopf equation for the surface boundary-layer. Numerical results are given for various choices of the thermal source and of the scattering kernel. Comparisons are made between the exact solution, for fixed but small ∊, and the leading terms of the asymptotic expansions. The surface boundary-layer solution always constitutes a very good representation (less than 1 per cent error, for ∊ < 10⁻⁴ and τ up to 10). The interior solution is also faithful for algebraically decreasing kernels (error less than 1 per cent for τ > 100). However, the interior solution constitutes a rather poor approximation when logarithmic terms are present (e.g. Doppler profiles