Anisotropic multiple scattering of collimated irradiance

Spherical harmonics are often used to solve multi-scatter transport problems. For a collimated incident beam, we describe a technique similar to the familiar P_N approximation except that we manage the infinite set of coupled equations in a novel way that permits the use of hundreds of harmonics to represent very pointed angular distributions. We split some equations into two approximate ones that depend on the angle of incidence. For plane slab geometry, the harmonics of low degree are solved in a coordinate system aligned to the boundary while the high harmonics use coordinates pointed at the source. We illustrate the technique with an extreme example: a scattering function having an integrable singularity in the forward direction. We give a time-dependent example along with several time-independent ones.