The particle distribution :math:`\psi(r, \Omega, E, t)` here is often named as angular flux, :math:`\{\Sigma_s, \Sigma_t \}` are the scattering and total cross sections correspondingly, and :math:`Q` denotes a source term.

The spherical harmonics moment equations

---------

The spherical harmonics (:math:`P_N`) method (cf. Brunner and Holloway [2005]) is one of several ways to solve the equation of radiative transfer.

It serves as an approximate method, i.e. the method of moments, to reduce the high dimensionality when the original kinetic equation of radiative transfer, which is formulated on a seven-dimensional domain.

Let us consider the radiative transfer equation in one-dimensional physical space with only one energy group, i.e.

Two properties of the spherical harmonics are crucial for our method. These appear here as properties of the Legendre polynomials. First, we observe that, by this

procedure, we have diagonalized the scattering operator on the right-hand side (the

Legendre polynomials are the eigenfunctions of scattering operator).

Second, a general property of orthogonal polynomials is that they satisfy a recursion relation. In