Impact of the numerical conversion to optical depth on the transfer of polarized radiation

Context. Making the conversion from the geometrical spatial scale to the optical depth spatial scale is useful in obtaining numerical solutions for the radiative transfer equation. This is because it allows for the use of exponential integrators, while enforcing numerical stability. Such a conversion involves the integration of the total opacity of the medium along the considered ray path. This is usually approximated by applying a piecewise quadrature in each spatial cell of the discretized medium. However, a rigorous analysis of this numerical step is lacking. Aims. This work is aimed at clearly assessing the performance of different optical depth conversion schemes with respect to the solution of the radiative transfer problem for polarized radiation, out of the local thermodynamic equilibrium. Methods. We analyzed different optical depth conversion schemes and their combinations with common formal solvers, both in terms of the rate of convergence as a function of the number of spatial points and the accuracy of the emergent Stokes profiles. The analysis was performed in a 1D semi-empirical model of the solar atmosphere, both in the absence and in the presence of a magnetic field. We solved the transfer problem of polarized radiation in different settings: the continuum, the photospheric Sr I line at 4607 Å modeled under the assumption of complete frequency redistribution, and the chromospheric Ca I line at 4227 Å, taking the partial frequency redistribution effects into account during the modeling. Results. High-order conversion schemes generally outperform low-order methods when a sufficiently high number of spatial grid points is considered. In the synthesis of the emergent Stokes profiles, the convergence rate, as a function of the number of spatial points, is impacted by both the conversion scheme and formal solver. The use of low-order conversion schemes significantly reduces the accuracy of high-order formal solvers. Conclusions. In practical applications, the use of low-order optical depth conversion schemes introduces large numerical errors in the formal solution. To fully exploit high-order formal solvers and obtain accurate synthetic emergent Stokes profiles, it is necessary to use high-order optical depth conversion schemes.