An Efficient Short Characteristic Solution for the Transfer Equation in Axisymmetric Geometries Using a Spherical Coordinate System

Abstract

We present a new formulation of the short characteristic method for solving the transfer equation in axially symmetric systems. The radiation field is solved for on a grid in r and the spherical polar angle β, with radiation coordinates chosen to take into account the spherical nature of the underlying source. Such a coordinate system is advantageous because it takes into account the inherent discontinuities and symmetries due to the forward-peaking nature of the radiation field. An important consequence of this is that the determination of the boundary condition for the intensity along a short characteristic does not involve an interpolation across a discontinuity. A parabolic approximation for the source function provides the best combination of accuracy, stability, and speed. Systematic errors introduced by using a weighted trapezoidal rule for quadrature in the radiation coordinate μ were overcome by using a monotonic cubic polynomial to approximate integrands. The code has been successfully used to solve the polarized transfer equation and employs both an approximate operator iteration and Ng acceleration to improve convergence. Comparison with an accurate long characteristic code shows that a tremendous saving in computation time can be realized with a minimal loss of accuracy.