An accurate and computationally efficient two-or three-dimensional (2-D or 3-D) neutron diffusion model will be needed to develop, compute safety parameters, and analyze the fuel cycle of a prismatic very high temperature reactor design under the Next Generation Nuclear Plant. To meet this need, an analytical Nodal Green's function method for the transverse integrated neutron diffusion equation was developed in 2-D and 3-D hexagonal geometry. This scheme is incorporated into the algorithm of the HEXPEDITE, a code first developed by Fitzpatrick and Ougouag. HEXPEDITE neglects discontinuous terms that arise in the transverse leakage because of the transverse integration procedure application to hexagonal geometry, and cannot account for the localized effects of burnable poisons across nodal boundaries. The test code being developed for this document accounts for these terms by maintaining a strict inventory of neutrons by using the nodal balance equation as a constraint of the neutron flux equation. The method developed in this report is intended to restore neutron conservation and increase the accuracy and fidelity of the code by adding the effect of the discontinuous terms to the transverse integrated flux solution. This is achieved through the use of the Nodal Green's function method to the complete equation resulting from the transverse integration procedure. The result is a rigorous semi-analytical solution. The new treatment of the burnable applies a similar approach, but with singular terms of pre-computed strength, based on blackness theory. vi vii ACKNOWLEDGEMENTS