Numerically Solving Radiative Transfer Equations

図: Top: Inhomogeneous point distribution representing a spiral galaxy and a logarithmic plot of the resulting radiation field from a central source. Bottom: Magnified view (16X) of the region within the white square.

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Ritzerveld, Icke and Rijkhorst developed a new numerical method for solving the equations of radiative transfer. It makes use of unstructured grids in which the grid-points are placed coincident with most of the optically active material, thus putting the accuracy where it is needed most.

This approach led them to several interesting and advantageous properties of the method. First of all, this method reduces the ordinary 7-dimensional set of equations to a 1-dimensional one, thus reducing the computational costs by enormous factors making it possible, for the first time, to do complicated multi-dimensional radiative transfer on a simple stand-alone computer and to couple it to existing hydodynamical schemes. Secondly, their use of unstructured, Lagrangian grids circumvents limitations commonly found in current transfer codes. For instance, these may be optimised only for axi-symmetric problems with a single central source, or while they may yield reliable solutions in one opacity regime, fail in others. Ritzerveld and coworkers could show that their method is as fast in every dimension, even when the number of sources increases and when the sources are not centrally distributed. They also showed that their method implicitly guarantees a correct treatment even when a photon propagates from an optically thick to an optically thin region, and the time-dependent transfer equations are solved implicitly not requiring extra computational effort.

図: Images of the newly detected ring systems in the halos of planetary nebulae. All images are in negative greyscale. The bottom row of images shows the original image in a logarithmic display, the middle row shows the processed images in a linear display, and the top row the processed images with circular fits of the rings.

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Subsequently, they used their SimpleX-code, a two- and three-dimensional implementation of their new method, to verify the method by testing several time-(in)dependent test-cases such as the creation of Strömgren spheres, the tracking of non-spherical ionisation fronts and various intrinsic three-dimensional inhomogeneous problems. An example is given in Fig.2.10 which shows a highly inhomogeneous matter distribution represented by an inhomogeneous grid-point distribution (top-left), and the result of the method for a central point source (top-right). The advantages of an unstructured, Lagrangian grid are highlighted by exemplifying the amount of detail with a zoom-in on the region within the white squares. The results are plotted in the bottom half of Fig.2.10