Linearizing Algorithm for Solving a Nonlinear Initial Boundary Value Problem of Radiation Transfer in Spatially Multidimensional Domains

The present work is devoted to the study of nonlinear integro-differential equations of the theory of radiation transfer and statistical equilibrium. Previously developed methods for solving the corresponding one-dimensional boundary value and initial boundary value problems are applied to problems of higher dimension. The issues of the existence and uniqueness of a non-negative solution of the corresponding nonlinear problems in bounded domains in spatially two-dimensional and three-dimensional cases are considered. The paper proposes and substantiates a linearizing iterative algorithm for solving nonlinear problems. The results of numerical implementation of this algorithm in the form of software modules within the framework of the finite element method and the finite difference method are presented. An integro-interpolation method is used for finite-difference approximation of the kinetic transfer equation, isoparametric elements are used in the finite element implementation. The efficiency and effectiveness of the proposed algorithm is numerically illustrated on model problems for specific media under various assumptions about the optical density of matter. The relations for obtaining Einstein coefficients in the media under consideration are discussed in detail.