Characteristic tube approach to analysis of DSn-method and to constructing new finite-difference schemes on Sn-nets

In this paper DSn-method for spherically symmetric transport equation is analyzed. It is shown to be split into a scheme for an ordinary differential equation (ODE) and a scheme for distributing the total flux over the unilluminated cell sides. The both are nonmonotone and nonpositive. Basing on the idea of splitting, a new approach to constructing finite-difference schemes for spherically symmetric transport equation on arbitrary nets is formulated and proved. In this approach the characteristic tubes conservative method is generalized on arbitrary nets, in part the Sn-nets. A new value—the full particle flux in the tube—is introduced. It gives an opportunity to pass from a partial differential equation (ODE) along 'average' characteristic in the Sn-cell. The average ODE obtained is treated as a balance equation of particles in the cell. The ODE is approximated by the monotony second order schemes. The full flux obtained is positive. It should be conservatively distributed oder the unilluminated Sn-cell sides in corresponding with additional approximation requirements. In the paper some variants of the full flux distribution are shortly discussed and the numerical calculations are given.