An arbitrarily high order transport method of the characteristic type for unstructured tetrahedral grids

An extension of the Arbitrarily High-Order Transport Method of the Characteristic type to three-dimensional unstructured tetrahedral grids is proposed in this work which resolves the difficulties encountered in the earlier formulation of the method. A thorough literature review is presented of the development of characteristic methods as a specific class of spatial discretization to the discrete-ordinates approximation of the stationary transport equation. A classical derivation of the Arbitrarily High-Order Transport Method of the Characteristic Type for Unstructured Grids (AHOT-C-UG) is performed, which is based on a consistent generalization of lower-order short characteristic methods in structured grids. Novel techniques, such as coordinate transformations and series expansions of spatial moments, are introduced in order to avoid previous difficulties regarding computational precision and the treatment of internal voids. In addition, the arbitrary-order characteristic relation is re-derived in a form which satisfies exactly the arbitrary-order balance equation. The consequences of the equivalence are twofold: it provides an exact relation that is numerically stable as the computational cells become optically thin, and it underscores the fact that the characteristic relation conserves the local balance of all computed spatial moments of the particle population, which is an important property of good spatial discretizations to the transport equation. Furthermore, the AHOT-C-UG approach is subsequently reintroduced as a Discontinuous Petrov-Galerkin projection, which allows us to bridge the gap between characteristic methods that are consistent with AHOT-C-UG and general finite element methods by providing a common setting from the point of view of variational analysis. Remaining mathematical and computational challenges regarding the AHOT-C-UG approach are noted and practical solutions to these issues are suggested. A rudimentary performance model of the method is introduced in order to address practical concerns regarding computational resources and runtime performance. Numerical results are shown which verify the validity of the performance model. In addition, the expected behavior of the method's convergence rate, based on the behavior of the error with respect to mesh refinement for a specified set of spatial expansion orders, is verified by performing a set of numerical experiments. Finally, a set of computational benchmarks are solved in order to show that the formalism can be used to analyze realistic radiation transport problems.