On the eigenstructure of spherical harmonic equations for radiative transport

The spherical harmonic equations for radiative transport are a linear, hyperbolic set of balance laws that describe the state of a system of particles as they advect through and collide with a material medium. For regimes in which the collisionality of the system is light to moderate, significant qualitative differences have been observed between solutions, based on whether the angular approximation used to derive the equations occurs in a subspace of even or odd degree. This difference can be traced back to the eigenstructure of the coefficient matrices in the advection operator of the hyperbolic system. In this paper, we use classical properties of the spherical harmonics to examine this structure. In particular, we show how elements in the null space of the coefficient matrices depend on the parity of the spherical harmonic approximation and we relate these results to observed differences in even and odd expansions.