Meshfree methods for nonlinear equilibrium radiation diffusion equation

Abstract

The radiation diffusion problem is usually applied to astrophysics and inertial confinement fusion. It seems that the traditional numerical methods such as finite difference method and finite element method is not efficiency to solve the problem because of its nonlinearity and complex computational domain. In this paper, some fast meshfree methods based on radial basis function are proposed to solve the equilibrium radiation diffusion equation. Because the modeling problem has two nonlinear terms, so we use the successive permutation iteration and linearization methods to approximate the nonlinear terms. Firstly, diffusion terms of the one-dimensional (1D) and two-dimensional (2D) equilibrium radiation diffusion equations are linearized by constructing a successive permutation iteration method for fully-implicit discretization. Then, three kinds of different linearization methods (factorization iteration, Picard–Newton iteration and Richtmyer iteration) are proposed and used to linearization of other nonlinear terms. Finally, the linearized equation is solved numerically by Kansa’s method with compactly supported radial basis function. The reliability, accuracy and effectiveness of new methods are verified by 1D and 2D numerical experiments. The advantages of presented method are apparent. They require neither domain nor boundary discretization, and no domain integration is required. New methods are easy for implementation and coding. They not only avoid the complexity of generation mesh, but also solve the equilibrium radiation diffusion equation with high accuracy.