Hybrid discrete ordinates and characteristics method for solving the linear Boltzmann equation

With the ability of computer hardware and software increasing rapidly, deterministic methods to solve the linear Boltzmann equation (LBE) have attracted some attention for computational applications in both the nuclear engineering and medical physics fields. Among various deterministic methods, the discrete ordinates method (SN) and the method of characteristics (MOC) are two of the most widely used methods. The SN method is the traditional approach to solve the LBE for its stability and efficiency. While the MOC has some advantages in treating complicated geometries. However, in 3-D problems requiring a dense discretization grid in phase space (i.e., a large number of spatial meshes, directions, or energy groups), both methods could suffer from the need for large amounts of memory and computation time.;In our study, we developed a new hybrid algorithm by combing the two methods into one code, TITAN. The hybrid approach is specifically designed for application to problems containing low scattering regions. A new serial 3-D time-independent transport code has been developed. Under the hybrid approach, the preferred method can be applied in different regions (blocks) within the same problem model. Since the characteristics method is numerically more efficient in low scattering media, the hybrid approach uses a block-oriented characteristics solver in low scattering regions, and a block-oriented SN solver in the remainder of the physical model.;In the TITAN code, a physical problem model is divided into a number of coarse meshes (blocks) in Cartesian geometry. Either the characteristics solver or the SN solver can be chosen to solve the LBE within a coarse mesh. A coarse mesh can be filled with fine meshes or characteristic rays depending on the solver assigned to the coarse mesh. Furthermore, with its object-oriented programming paradigm and layered code structure, TITAN allows different individual spatial meshing schemes and angular quadrature sets for each coarse mesh. Two quadrature types (level-symmetric and Legendre-Chebyshev quadrature) along with the ordinate splitting techniques (rectangular splitting and PN-TN splitting) are implemented. In the S N solver, we apply a memory-efficient 'front-line' style paradigm to handle the fine mesh interface fluxes. In the characteristics solver, we have developed a novel 'backward' ray-tracing approach, in which a bi-linear interpolation procedure is used on the incoming boundaries of a coarse mesh. A CPU-efficient scattering kernel is shared in both solvers within the source iteration scheme. Angular and spatial projection techniques are developed to transfer the angular fluxes on the interfaces of coarse meshes with different discretization grids.;The performance of the hybrid algorithm is tested in a number of benchmark problems in both nuclear engineering and medical physics fields. Among them are the Kobayashi benchmark problems and a computational tomography (CT) device model. We also developed an extra sweep procedure with the fictitious quadrature technique to calculate angular fluxes along directions of interest. The technique is applied in a single photon emission computed tomography (SPECT) phantom model to simulate the SPECT projection images. The accuracy and efficiency of the TITAN code are demonstrated in these benchmarks along with its scalability. A modified version of the characteristics solver is integrated in the PENTRAN code and tested within the parallel engine of PENTRAN. The limitations on the hybrid algorithm are also studied.