Phase-space finite element methods applied to the first-order form of the transport equation

Abstract

The application of the finite element method to the first-order form of the neutron transport equation is reviewed. The general theoretical foundation of the finite element application is summarized, including a derivation of the weak form, a discussion of the treatment of all boundary conditions as natural boundary conditions and a few remarks concerning convergence. Results of the 1-D application are presented including a description of the discontinuous phase-space finite elements. The 2-D application is discussed and its application to the classic ray effect problem is examined. It is concluded that the finite element method does alleviate the ray effect but at the considerable expense of computational time and memory requirements. To address this concern, a new `segmentation' scheme for the 2-D application is described. This scheme yields satisfactory results for the ray effect problem while reducing the computational cost by nearly an order of magnitude. Finally a few remarks are presented concerning the time-dependent application and the paper concludes with some general comments concerning the overall application of the finite element method to the first-order equation and comparison with alternative methods.