A New Fourth-Order Finite-Difference Method for Solving Discrete-Ordinates Slab Transport Equations

Abstract

This work is concerned with a theoretical study of a new fourth-order finite-difference scheme for spatially discretizing the discrete-ordinates equations for solving numerically the slab transport (Boltzmann) equation. This analysis considers the quadratic continuous method, whose derivation parallels that of the commonly used diamond difference and linear discontinuous schemes from balance equations for particle conservation across a spatial cell. We provide a convergence analysis of this method and prove that superconvergence phenomena are present for cell-edge and cell-average fluxes. We also present results from an S2-test problem to show that the asymptotic convergence rates are observed on rather coarse spatial meshes.

On superconvergence techniques

“…An averaging technique for the Stokes problem can be found in Johnson and Pitk~iranta [104]. We also mention the paper [142] of Neta and Victory, where superconvergence phenomena have been presented for cell-edge and cell-average fluxes. We also mention the paper [142] of Neta and Victory, where superconvergence phenomena have been presented for cell-edge and cell-average fluxes.…”