Numerical solution of linear transport equations with scattering operators of integral and differential type

We revisit the PN-method and demonstrate its ability to provide accurate and timely solutions to transport equations with scattering operators whose only restriction is having spherical harmonics as eigenfunctions. Such operators include the integral scattering operator characterized by a scattering kernel and forward-scattering approximations thereto involving functions of the Laplacian restricted to the unit sphere. Solutions obtained with the PN-method are shown to converge exponentially as the number, N, of Legendre polynomials used to approximate the solution is increased. The computation time is shown to grow slowly with increased number of polynomials used. Moreover the solution technique is shown to be stable as N increases. The primary result of this work is the use of the PN-method to carry out prompt side-by-side comparisons of radiative transport equations with different tissue-light interactions. Comment is made on the ability of the forward-scattering approximations to describe the transport properties of light in biological media as determined by the radiative transfer equation with Henyey-Greenstein phase function.