Diffusive corrections to $P_N$ approximations

In this paper, we investigate moment methods from a general point of view using an operator notation. This theoretical approach lets us explore the moment closure problem in more detail. This gives rise to a new idea, proposed in Levermore [Transition Regime Models for Radiative Transport, Presentation at the Institute for Pure & Applied Mathematics: Grand Challenge Problems in Computational Astrophysics, Workshop on Transfer Phenomena, 2005, http://www.ipam.ucla.edu/publications/pcaws4/pcaws_45724.pdf, and Moment Closures for Radiative Transport, manuscript, 2009], of how to improve the well-known P-N approximations. We systematically develop a diffusive correction to the PN equations from the operator formulation-the so-called D-N approximation. We validate the new approach with numerical examples in one and two dimensions.