High-precision meshless method for 3D radiation diffusion problem in sphere and cylinder

The problem of radiation diffusion is extremely challenging due to the complex physical processes and nonlinear characteristics of the equation involved. In this paper, we propose a class of high-precision meshless methods for 3D nonlinear radiation diffusion equations applicable to spherical and cylindrical walls. Firstly, when the energy density is linearly related to temperature, we use a full-implicit difference scheme to discretize the time term, and then approximate the spatial term using radial basis functions to construct a new solution scheme for solving the 3D linear radiation diffusion equation. Secondly, when dealing with the nonlinear relationship between energy density and temperature, we successfully reduced the complexity of problem to be by linearizing $T^4$. Then, we use radial basis functions to approximate unknown functions and established a large class of solving schemes, which solved by the Kansa’s method. Finally, we validate the efficiency and high accuracy of the proposed methods through a series of numerical examples on spherical and cylindrical walls. In summary, the meshless numerical solution methods proposed in this paper not only avoids the complexity of meshing in irregular areas, but also provides a new and high-precision numerical solution method for the 3D radiation diffusion equation.