On radiative transfer using synthetic kernel and simplified spherical harmonics methods in linearly anisotropically scattering media

•SKN methods are extended to linear anisotropic scattering media.•Diagonalized simplified PN and simplified DPN equations are also derived.•PN and DPN quadratures are used with the SKN and SKN⁎ methods.•SKN methods are equivalent to SP2N−1 and SDPN−1 approximations.

The Synthetic Kernel (SKN) method is employed to a 3D absorbing, emitting and linearly anisotropically scattering inhomogeneous medium. Standard SKN approximation is applied only to the diffusive components of the radiative transfer equations. An alternative SKN (SKN⁎) method is also derived in full 3-D generality by extending the approximation to the direct wall contributions. Complete sets of boundary conditions for both SKN approaches are rigorously obtained. The simplified spherical harmonics (P2N−1 or SP2N−1) and simplified double spherical harmonics (DPN−1 or SDPN−1) equations for linearly anisotropically scattering homogeneous medium are also derived. Resulting full P2N−1 and DPN−1 (or SP2N−1 and SDPN−1) equations are cast as diagonalized second order coupled diffusion-like equations. By this analysis, it is shown that the SKN method is a high-order approximation, and simply by the selection of full or half range Gauss-Legendre quadratures, SKN⁎ equations become identical to P2N−1 or DPN−1 (or SP2N−1 or SDPN−1) equations. Numerical verification of all methods presented is carried out using a 1D participating isotropic slab medium. The SKN method proves to be more accurate than SKN⁎ approximation, but it is analytically more involved. It is shown that the SKN⁎ with proposed BCs converges with increasing order of approximation, and the BCs are applicable to SPN or SDPN methods.