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Photospheric radiation can drive winds from hot, massive stars by direct momentum transfer through scattering in bound-bound transitions of atmospheric ions. The line radiation force should cause a new radiative wave mode. The dispersion relation from perturbations of the line force was analysed so far either in Sobolev approximation or for pure line absorption. The former does not include the line-driven instability, and the latter cannot account for upstream propagating, radiative waves. We consider a non-Sobolev line force that includes scattering in a simplified way, accounting however for the important line-drag effect. We derive a new dispersion relation for radiative waves, and analyse wave propagation using Fourier methods, and by numerical solution of an integro-differential equation. The existence of an upstream propagating, dispersive radiative wave mode is demonstrated.
We present a new numerical algorithm to solve the recently derived equations of two-moment cosmic ray hydrodynamics (CRHD). The algorithm is implemented as a module in the moving mesh AREPO code. Therein, the anisotropic transport of cosmic rays (CRs) along magnetic field lines is discretized using a path-conservative finite volume method on the unstructured time-dependent Voronoi mesh of AREPO. The interaction of CRs and gyroresonant Alfven waves is described by short time-scale source terms in the CRHD equations. We employ a custom-made semi-implicit adaptive time stepping source term integrator to accurately integrate this interaction on the small light-crossing time of the anisotropic transport step. Both the transport and the source term integration step are separated from the evolution of the magnetohydrodynamical equations using an operator split approach. The new algorithm is tested with a variety of test problems, including shock tubes, a perpendicular magnetized discontinuity, the hydrodynamic response to a CR overpressure, CR acceleration of a warm cloud, and a CR blast wave, which demonstrate that the coupling between CR and magnetohydrodynamics is robust and accurate. We demonstrate the numerical convergence of the presented scheme using new linear and non-linear analytic solutions.