Application of the smoothed particle hydrodynamics method to the neutron diffusion equation

A novel meshless method, known as Smoothed Particle Hydrodynamics (SPH), is applied to solve the neutron diffusion equation for the first time. The discrete forms for each term of the neutron diffusion equation and the boundary conditions are presented, along with a discussion on the differences among various discretization strategies. A comprehensive workflow for solving the neutron diffusion equation using SPH is provided, and the properties of several commonly used kernel functions are examined. An in-house code was developed, and the research indicates that not discretizing the constant terms in the diffusion equation but instead using the values from the previous iteration results in higher accuracy and is less sensitive to variations in parameters such as the smoothing length. A similar finding is observed for boundary conditions; when the boundary condition is of the first kind, directly assigning values yields greater accuracy than using kernel function interpolation. Among the compared kernel functions, the super-Gaussian kernel exhibits the best performance, with errors not exceeding 2.38 % under the given conditions, significantly outperforming the Gaussian and quintic spline kernel functions.