Localized Radial Basis Functions for Radiation Transport in Slab Geometry

Meshing the spatial domain for the neutron transport equation can be computationally expensive and labor-intensive. Mesh-free methods, such as radial basis function (RBF) approximations, may alleviate these concerns by using a pointwise solution of the transport equation instead of the standard volume-integrated solution using cells or elements. This allows for the use of any geometric description that can provide the material at solution points and the boundary normal directions for boundary points. RBFs give spectral convergence when used to solve differential equations, but only if they are infinitely smooth. This occurs when the RBFs are effectively flat, creating an ill-conditioned, dense system of equations. One simple way to alleviate this issue is to use localized RBFs. Global RBFs have been applied previously to the monoenergetic radiative transport equation in X-Y geometry using source iteration. In this summary, localized RBFs are applied to the multigroup neutron transport equation in slab geometry using Krylov iteration. The results show agreement with benchmark solutions, provided the coefficients for the RBF basis are properly chosen. (authors)