The Method of Spherical Harmonics (PN-Approximation)

Abstract

In this chapter, the method of spherical harmonics ( P N -approximation) is applied to address the directional dependence of the intensity in the radiative transfer equation (RTE). The approximation is first applied to a one-dimensional slab and the resulting differential equations and boundary conditions are developed. The RTE with the lowest-order P 1 -approximation is then formulated for multidimensional media, followed by development of the P 3 and higher-order approximations. Both Cartesian and cylindrical coordinate systems are considered. Simplification to the P N -approximation—the so-called S P N approximation—is presented next. Several enhancements to the basic P 1 -approximation (or Ordinary Differential Approximation, ODA) are discussed. These include the Advanced Differential Approximation (ADA), the Improved Differential Approximation (IDA), and the Modified Differential Approximation (MDA). Finally, the accuracy and efficiency of the various methods are compared.