On the Accuracy and Stability of the Meshless RBF Collocation Method for Neutron Diffusion Calculations

Abstract

Accuracy and stability are the main properties that make an algorithm preferable to its counterparts in modelling of physical phenomena. The radial basis function collocation method is a novel meshless technique, which exhibits an exponential convergence rate for the numerical solution of partial differential equations. However, it is a global approximation scheme and the ill-conditioning of the collocation matrix may become a serious issue if dense sets of interpolation nodes or high values of shape parameters are utilized. This study discusses four strategies to improve the accuracy and stability of the radial basis function collocation method for the numerical solution of the multigroup neutron diffusion equation. These strategies include using a higher precision value for computations, utilizing higher exponents for the generalized multiquadric, decreasing the value of the shape parameter with the number of nodes and singular value decomposition filtering. The results have shown that by using a higher precision value, choosing a variable shape parameter strategy and filtering the smallest singular values of the collocation matrix it is possible to improve the performance of the meshless collocation method, while increasing the exponent of the multiquadric results in a more accurate but less stable algorithm.