An analysis of the finite-differenced, even-parity discrete-ordinates equations in slab geometry

Considerable effort has been expended in recent years in finding improved spatial differencing schemes for the neutron and radiation transport equations. Standard criteria used to select a candidate scheme are its order of spatial convergence for small mesh size and its positivity in the sense that positive solutions emerge from positive input data. More recently, it has become clear that truly robust schemes must behave well in diffusing regions and must be compatible with an effective iteration acceleration method. Recently, Morel and Larsen reported their work on a promising new method called the multiple balance method that has virtually all the desirable characteristics. Here we study a different approach to the problem by considering discrete-ordinates approximations to the even-parity transport equations. We analyze three spatial difference approaches: diamond differencing, cell-edge differencing, and cell-center differencing. For the case of isotropic scattering and sources, the latter two approaches are shown to be strictly positive, to be second-order accurate, to be compatible with derived diffusion synthetic acceleration methods, and to possess the necessary diffusion limits. Unlike previous work with the even-parity equation, we do not use finite elements or variational principles. 5 refs., 1 tab