| Abstract
| - Context. Observations of the solar atmosphere of ever increasing spatial resolution reveal steep gradients in the magnetic field and in thermal states. Likewise, numerical simulations of the solar atmosphere show contact discontinuities and shock fronts. This asks for the development of robust methods for computing the radiative transfer of polarized light in discontinuous media. Aims. Here, we propose a new concept for dealing with discontinuities in the radiative transfer of polarized light and carry out a few basic test calculations. While in the past, the focus was on interpolating the source function with ever-increasing accuracy and smoothness, we propose to take the opposite approach by reconstructing it with piecewise continuous functions, taking discontinuities on purpose into account. This concept is known from computational fluid dynamics. Methods. Test calculations were carried out for (i) a Milne-Eddington atmosphere; (ii) an atmosphere featuring a single discontinuity that is shifted across one grid cell; and (iii) a two-layered atmosphere with discontinuities in the source function, the velocity, and the magnetic field. Results. It is shown that the method of piecewise continuous reconstruction is a viable approach to solving the radiative transfer equation for polarized light. In the special case where a discontinuity coincides with a computational cell interface, the method is capable of producing the exact solution. Overall, the assessment of the piecewise continuous reconstruction method turns out to be cautiously positive, but it does not lead to an order-of-magnitude improvement in accuracy over conventional methods for the examples considered here. More realistic model atmospheres need to be considered for judging practical applicability.
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