Abstract
A nonlinear system of integro-differential equations of radiation transfer and statistical equilibrium in a plane-parallel layer is studied within the framework of a two-level atom model under the assumption of a complete redistribution of radiation in frequency. A boundary value problem for the kinetic transport equation with a condition corresponding to the absence of an external particle flux incident on the boundary of the region is considered. The results on the existence and uniqueness of the solution of the problem are presented. To find this solution, an iterative linearizing algorithm is proposed and justified. A finite-difference scheme of the integro-interpolation method is used for the numerical solution of the problem. Its main properties - the stability condition, the approximation order, the conservativeness condition of the scheme - are investigated. The efficiency of the algorithm is numerically illustrated on model problems for specific media under various assumptions about the optical density of the matter.
Supported by the Scientific and Education Mathematical Center “Mathematics for Future Technologies” (Project No. 075-02-2022-883).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Vladimirov, V.S.: Mathematical problems in the one-velocity theory of particle transport. Tr. MIAN SSSR 61. Publ. House of the Academy of Sciences of the USSR, Moscow (1961)
Germogenova, T.A.: Local Properties of the Transport Equation Solutions. Nauka, Moscow (1986). (in Russian)
Marchuk, G.I., Lebedev, V.I.: Numerical Methods in the Theory of Neutron Transport. Harwood Academic Pub (1986)
Prokhorov, I.V., Sushchenko, A.A.: The Cauchy problem for the radiative transfer equation in an unbounded medium (in Russian). Dal’nevost. Mat. Zh. 18(1), 101–111 (2018). https://doi.org/10.1134/S0001433821030105
Amosov, A.A.: Initial-boundary value problem for the nonstationary radiative transfer equation with diffuse reflection and refraction conditions. J. Math. Sci. 235(2), 117–137 (2018). https://doi.org/10.1007/s10958-018-4063-y
Sushkevich, T.A., Falaleeva, V.A.: Hyperspectral model of solar radiation transfer in an atmosphere with cirrus clouds Izv. Atmos. Ocean. Phys. 57(3), 277–285 (2021)
Bass, L.P., Voloshchenko, A.M., Germogenova, T.A.: Methods of discrete ordinates in problems of radiation transfer. IPM im. Keldysha AN SSSR, Moscow (1986). (in Russian)
Gol’din, V.Ya.: A quasi-diffusion method of solving the kinetic equation. Comp. Math. Math. Phys. 4(6), 136–149 (1964). https://doi.org/10.1016/0041-5553(64)90085-0
Ivanov, V.V.: Transfer Theory and the Spectra of Calestial Objects. Nauka, Moscow (1969). (in Russian)
Mihalas, D.: Stellar Atmospheres. Freeman, San Francisco (1978)
Chetverushkin, B.N., Mingalev, I.V., Chechetkin, V.M., Orlov, K.G., Fedotova, E.A., Mingalev, V.S.: Block of calculation of the solar radiation field in the general circulation model of the Earth’s lower and middle atmosphere (in Russian). Matem. Mod. 34(3), 43–70 (2022). https://doi.org/10.20948/mm-2022-03-03
Kalinin, A.V., Morozov, S.F.: A non-linear boundary-value problem in the theory of radiation transfer. Comput. Math. Math. Phys. 30(4), 76–83 (1990). https://doi.org/10.1016/0041-5553(90)90046-U
Kalinin, A.V., Morozov, S.F.: Solvability “in the large’’ of a nonlinear problem of radiative transfer. Differ. Uravn. 21(3), 484–494 (1985)
Kalinin, A.V., Morozov, S.F.: A mixed problem for a nonstationary system of nonlinear integro-differential equations. Sib. Math. J. 40(5), 887–900 (1999). https://doi.org/10.1007/BF02674718
Kalinin, A.V., Tyukhtina, A.A.: On a nonlinear problem for a system of integro-differential equations of radiative transfer theory. Comput. Math. Math. Phys. 62(6), 933–944 (2022). https://doi.org/10.1134/S0965542522060094
Kalinin, A.V., Kozlov, A.Y.: On anonlinear problem of radiative transfer in a plane-parallel layer (in Russian). Vest. NNGU 4(1), 140–145 (2011)
Birkhoff, G.: Lattice Theory. American Mathematical Society (1961)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press (1982)
Kalitkin, N.N.: Numerical Methods. Nauka, Moscow (1978). (in Russian)
Frisch, S.E.: Optical Spectra of Atoms. Fizmatlit, Moscow (1963). (in Russian)
Elyashevich, M.A.: Atomic and Molecular Spectroscopy. Fizmatlit, Moscow (1962). (in Russian)
Allen, C.W.: Astrophysical Quantities. The Athlone Press, London (1973)
Zel’dovich, Ya.B., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, Cambridge (1967)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Kalinin, A., Tyukhtina, A., Busalov, A. (2022). An Iterative Method for Solving a Nonlinear System of the Theory of Radiation Transfer and Statistical Equilibrium in a Plane-Parallel Layer. In: Balandin, D., Barkalov, K., Meyerov, I. (eds) Mathematical Modeling and Supercomputer Technologies. MMST 2022. Communications in Computer and Information Science, vol 1750. Springer, Cham. https://doi.org/10.1007/978-3-031-24145-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-24145-1_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-24144-4
Online ISBN: 978-3-031-24145-1
eBook Packages: Computer ScienceComputer Science (R0)