In this paper, two new subcell balance methods for solving the multigroup discrete ordinates transport equation in unstructured geometrical problems are presented. The problem domains are divided into tetrahedral meshes to model the complicated geometries. In these new methods, the angular flux and its flux moments are approximated with the four-term linear discontinuous expansion, and then, the unknowns (four point fluxes or subcell average fluxes) and the interface fluxes are represented in terms of the expansion coefficients. Finally, the external and internal interface average fluxes are represented in terms of the unknown fluxes, and the subcell balance equations give the complete relations associated with the unknown fluxes.

Two ways for dividing a tetrahedral mesh into subcells are considered, and they lead to the new methods. The first subcell balance method, called LDEM-SCB(0), is relatively simple, and the second subcell balance method, called LDEM-SCB(1), is more complicated than LDEM-SCB(0). The point flux formulations of these methods can be easily implemented with minor modifications in the discontinuous finite element method codes. The numerical tests show that the new subcell balance methods provide accurate and robust solutions. In particular, the numerical analysis shows that LDEM-SCB(0) and LDEM-SCB(1) have first- and second-order accuracies, respectively, in the transport regime. Also, it was found from the asymptotic analysis that these methods satisfy the linear continuous diffusion discretizations on the interior in the thick diffusion limit.