Method of long characteristics applied in space and time

We have developed a long-characteristic (LC) discretization of the time and space variables in the transport equation and tested it in slab geometry. The method's sole approximation is in the spatial shape of the scattering-plus- fission source. Many LC spatial discretizations assume constant sources in each cell, but because we are interested in thick diffusive problems we must construct sources with at least linear variation in each cell. We have developed a least-squares procedure for constructing such sources. We recognize that it is not possible to simultaneously obtain all three of the following desirable properties: 1) exact solution along each ray in the purely-absorbing limit; 2) cellwise particle conservation; 3) smoothly varying cellwise reaction rates in smooth problems. To illustrate this we construct and display a cellwise-linear scalar flux that produces conservative reaction rates and one that produces smooth reaction rates. These quantities are similar for most cases and the difference between them vanishes in the limit of fine ray spacing. We compare our LC results against results from a traditional linear discontinuous spatial discretization with standard finite-difference time discretizations. We find that our method is more accurate for both streaming-dominated and scattering-dominated test problems. Finally, we remark that application of this method in parallel looks promising, partly due to the independence of the separate rays along which the solution is computed.