Relationship of the SPN_AN method to the even-parity transport equation

There are two distinct aspects to this paper. Firstly, it is shown that the AN form of the SPN equations can be derived from a suitably discretised form of the even parity transport equation. The procedure for doing this is explained and some limitations of SPN_AN theory are illustrated. Secondly, to show the magnitude of the errors in AN theory we have taken the problem of an infinitely repeating lattice in which the cell structure is symmetric but can contain any number of sub-regions. The regions can be elliptical and/or rectangular. Because of the repeating lattice, we can express the flux in the cell in the form of a Fourier series. The expansion coefficients of the series may then be obtained from a set of linear equations. It is found that the series converges very slowly and a large number of terms is required which can significantly increase the computational time required for accurate solutions. Use is made of a high order SN (N = 200) method to compare and test convergence of the Fourier method for an exact case and for the SPN equivalent.