Geodesic
Two approaches to effectively reduce the size of radiative heat transfer problems in multidimensional geometry
Matematičeskoe modelirovanie, Tome 33 (2021) no. 9, pp. 35-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

Mathematical modelling of kinetic nonstationary transfer of radiation energy is a laborintensive and consuming task. This is due to nonlinearity and high dimensionality of the system to solve. Generaly the kinetic transfer equation is solved in 7-dimensional phase space, which requires vast computational resources. Historically, some attempts were made to simplify the initial system to solve. But simplifying assumptions can a priori degrade the solution quality. Quasi-diffuse approximation for neutron transfer proposed in 1964 by V.Ya. Gol'din was a meaningful step forward in this direction and later it became one of the effective methods to solve uncharged particle transfer problems. The quasi-diffusion method makes allowance for kinetic effects via coefficients calculated in periodic solving of the kinetic equation. There exist other approaches to simplify the initial system. In 2010 M.Yu. Kozmanov and N.G. Karlyhanov proposed a model for 1D geometry which was ideologically close to the quasi-diffusion algorithm. The coefficients obtained in solving the kinetic equation are entered into the model. The approach is being actively developed at RFNC-VNIITF on practical as well as on theoretical level and the user experience suggests its wide application. The paper briefly describes two models and set out the calculation results for two test problems in 2D axially symmetric geometry.
Keywords: radiative heat transfer, dimensionality reduction, numerical method. 
@article{MM_2021_33_9_a2,
     author = {S. A. Grabovenskaya and V. V. Zaviyalov and A. A. Shestakov},
     title = {Two approaches to effectively reduce the size of radiative heat transfer problems in multidimensional geometry},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {35--46},
     publisher = {mathdoc},
     volume = {33},
     number = {9},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2021_33_9_a2/}
}
TY  - JOUR
AU  - S. A. Grabovenskaya
AU  - V. V. Zaviyalov
AU  - A. A. Shestakov
TI  - Two approaches to effectively reduce the size of radiative heat transfer problems in multidimensional geometry
JO  - Matematičeskoe modelirovanie
PY  - 2021
SP  - 35
EP  - 46
VL  - 33
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2021_33_9_a2/
LA  - ru
ID  - MM_2021_33_9_a2
ER  - 
%0 Journal Article
%A S. A. Grabovenskaya
%A V. V. Zaviyalov
%A A. A. Shestakov
%T Two approaches to effectively reduce the size of radiative heat transfer problems in multidimensional geometry
%J Matematičeskoe modelirovanie
%D 2021
%P 35-46
%V 33
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2021_33_9_a2/
%G ru
%F MM_2021_33_9_a2
S. A. Grabovenskaya; V. V. Zaviyalov; A. A. Shestakov. Two approaches to effectively reduce the size of radiative heat transfer problems in multidimensional geometry. Matematičeskoe modelirovanie, Tome 33 (2021) no. 9, pp. 35-46. http://geodesic.mathdoc.fr/item/MM_2021_33_9_a2/

[1] M. Necati Ozisik, Radiative Transfer and Interactions with Conduction and Convection, John Wiley Sons Inc, 1973

[2] B. N. Chetverushkin, Matematicheskoe modelirovanie zadach dinamiki izluchaiushchego gaza, Nauka, M., 1985

[3] V. Ya. Gol'din, “A quasi-diffusion method of solving the kinetic equation”, Computational Math. and Mathematical Physics, 4:6 (1964), 136–149 | DOI

[4] A. S. Eddington, The Internal Constitution of Stars, Cambridge University Press, 1926 | MR | Zbl

[5] D. Yu. Anistratov, E. N. Aristova, V. Ya. Gol'din, “Nelineinyi metod resheniia zadach perenosa izlucheniia v srede”, Matematicheskoe modelirovanie, 8:12 (1996), 3–28 | MR | Zbl

[6] E. N. Aristova, V. Ya. Gol'din, A. V. Kolpakov, “Metodika rascheta perenosa izlucheniia v tele vrashcheniia”, Matematicheskoe modelirovanie, 9:3 (1997), 91–108 | MR | Zbl

[7] E. N. Aristova, V. Ya. Gol'din, “Low_cost Calculation of Multigroup Neutron Transport Equation for the Recalculation of Spectrum_Averaged Cross Sections”, Mathematical Models and Computer Simulations, 1:5 (2009), 561–572 | DOI | Zbl

[8] D. Y. Anistratov, V. Ya. Gol'din, “Multilevel Quasi-diffusion Methods for Solving Multi-group Neutron Transport k-Eigenvalue Problems in One-Dimensional Slab Geometry”, Nuclear Science and Engineering, 169 (2011), 111–132 | DOI

[9] G. V. Dolgoleva, “Metodika rascheta dvizheniia dvukhtemperaturnogo izluchaiushchego gaza (SND)”, VANT, seriia «Matematicheskoe modelirovanie fizicheskikh protsessov», 1983, no. 2(13), 29–33

[10] N. G. Karlyhanov, M. Yu. Kozmanov, “Uchet kineticheskikh effektov v diffuzionnom priblizhenii dlia rascheta perenosa izlucheniia”, VANT, seriia «Matematicheskoe modelirovanie fizicheskikh protsessov», 2010, no. 4, 3–9

[11] A. A. Shestakov, “Study of Various Approximations Used in Modeling Radiative Heat Transfer Problems”, Math. Models Computer Simul., 13:2 (2021), 231–243 | DOI | MR | Zbl

[12] A. A. Shestakov, “Analysis of the Quasi-Transfer Approximation in Problems with Analytical Solution”, Computational Math. and Mathematical Physics, 60:5 (2020), 833–843 | DOI | MR | Zbl

[13] A. D. Gadzhiev, V. V. Zaviyalov, A. A. Shestakov, “Primenenie TVD podkhoda k DS$_n$ metodu resheniia uravneniia perenosa teplovogo izlucheniia”, VANT, seriia «Matematicheskoe modelirovanie fizicheskikh protsessov», 2009, no. 2, 37–48

[14] A. D. Gadzhiev, V. V. Zaviyalov, A. A. Shestakov, “Primenenie TVD podkhoda k DS$_n$ metodu resheniia uravneniia perenosa teplovogo izlucheniia v osesimmetrichnoi RZ geometrii”, VANT, seriia «Matem. modelirovanie fizicheskikh protsessov», 2010, no. 2, 30–39

[15] S. A. Grabovenskaya, A. A. Shestakov, “Analiz nekotorykh skhem dlia resheniia uravneniia perenosa izlucheniia kvazidiffuzionnym metodom”, VANT, seriia «Matematicheskoe modelirovanie fizicheskikh protsessov», 2011, no. 4, 3–15

[16] S. A. Grabovenskaya, V. V. Zaviyalov, A. A. Shestakov, “Konechno-raznostnaia skhema GROM dlia resheniia dvumernoi kvazidiffuzionnnoi sistemy perenosa teplovogo izlucheniia”, VANT, seriia «Matem. modelirovanie fizicheskikh protsessov», 2014, no. 3, 47–58

[17] A. D. Gadzhiev, V. N. Seleznev, E. M. Romanova, A. A. Shestakov, “Metodika TOM4-KD dlia matematicheskogo modelirovaniia dvumernykh uravnenii perenosa izlucheniia v mnogogruppovom kvazidiffuzionnom priblizhenii”, VANT, seriia «Matematicheskoe modelirovanie fizicheskikh protsessov», 2001, no. 4, 48–59

[18] A. D. Gadzhiev, A. A. Shestakov, “Metod vydeleniia diagonalnoi matritsy dlia chislennogo resheniia uravneniia perenosa izlucheniia v P$_1$-priblizhenii po skheme ROMB”, VANT, seriia «Matematicheskoe modelirovanie fizicheskikh protsessov», 2006, no. 1, 3–13

[19] J. A. Fleck, J. D. Cummings, “An Implicit Monte-Carlo Scheme for Calculating Time and Frequency Dependent Nonlinear Radiation Transport”, J. C. P., 8:3 (1971), 313–342 | MR | Zbl

[20] Yu. V. Yanilkin, Yu. A. Bondarenko, E. A. Goncharov, A. R. Guzhova, V. Yu. Kolobianin, V. N. Sofronov, V. P. Statsenko, Testy dlia gidrokodov, modeliruiushchikh udarnovolnovye techeniia v mnogokomponentnykh sredakh, v. 1, FGUP «RFIATS-VNIIEF», Sarov, 2017