On Finite-Difference Methods for Solving Discrete-Ordinates Transport Equations

Abstract

A unified error analysis is carried out for two general classes of finite-difference methods for solving discrete-ordinates slab transport equations. Our treatment includes, as special cases, the analysis by E. Larsen and P. Nelson [SIAM J. Numer. Anal., 19 (1982), pp. 334–348] and by B. Neta and H. D. Victory, Jr. [Numer. Funct. Anal. and Optimz., 5(1) (1982), pp. 85–126; SIAM J. Numer. Anal., 20 (1983), pp. 94–105] on several specific finite-difference methods which are linear in nature. In our analysis, we show how the finite-difference approximates at the mesh points can be naturally extended to yield approximates for all values of the spatial variable and obtain a priori estimates of the global discretization errors for these approximates. In addition we show that superconvergence occurs for the (original) finite-difference approximates to the exact discrete-ordinates angular flux at the mesh points and for approximates to several of its spatial moments defined on each mesh cell. We present computational results which compare the numerical behavior of some of the higher order methods with that of several of the methods discussed in the above cited works.