Pebble-bed high-temperature gas-cooled reactors (HTGRs) employ dispersed spherical fuel elements, where the diffusion of fission products within the fuel matrix and their subsequent release at the fuel surface play a critical role in radioactive safety analysis. The FRESCO-II code, known for its conservative modeling approach, is widely used to simulate these diffusion and release processes. However, its reliance on fully-implicit Euler time discretization and the finite volume method often leads to reduced accuracy, particularly at low temperatures or during the early stages of the simulation, resulting in numerical oscillations. This study presents improved numerical methods for solving the nuclide diffusion equation, employing finite difference methods (FDM), and meshing bias. These methods are specifically tailored for intact particles, failed particles, and graphite matrix, with distinct approaches for each sphere type. Comparative simulations against the FRESCO-II code and analytical solutions demonstrate that the proposed methods significantly mitigate numerical oscillations and enhanced accuracy. When applied to the simulation of fission product diffusion in HTR-PM fuel elements, the results confirm that the new methods achieve both computational efficiency and precision, offering a robust solution for modeling diffusion behavior in spherical fuel elements.