Parallel FE approximation of the even/odd-parity form of the linear Boltzmann equation

Abstract

A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional multigroup discreteordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the spacedirection dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.

Spatial discretizations for self-adjoint forms of the radiative transfer equations

“…(A.15) and (A. 16) in Appendix A, were used in the SAAI calculations for both discretizations. Unless otherwise stated, comparisons between the standard and alternative SAAI discretization are made in terms of the primary (vertex-centered) solutions.…”

“…The use of higher-order continuous finite-element spatial discretizations should be examined in the future to see how robustness is affected. One would generally expect higher-order methods to be even less robust than lower-order methods, but good results for multidimensional coupled electron-photon transport calculations have been obtained with continuous quadratic spatial elements [14,16].…”

“…However, the authors did not use the principle for this purpose, and made no reference to the SAAF equation. The EP, OP and SAAF equations have recently been applied to coupled electron-photon transport [14][15][16]. This application significantly differs from neutronics and radiative transfer applications because the small energy transfers associated with electron scattering generally requires special treatment.…”

“…This application significantly differs from neutronics and radiative transfer applications because the small energy transfers associated with electron scattering generally requires special treatment. In particular, these transfers can be treated either with a special type of ''condensed'' cross-section approximation [14,16], or a continuous-slowing-down term [15]. The former is compatible with standard discretization and solution techniques for the EP, OP, and SAAF equations, while the latter requires some generalization of these techniques.…”