Explicit-implicit difference scheme for the joint solution of the radiative transfer and energy equations by the splitting method

N. Ya. Moiseev

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 3, pp. 442-458

Résumé

High-order accurate explicit and implicit conservative predictor-corrector schemes are presented for the radiative transfer and energy equations in the multigroup kinetic approximation solved together by applying the splitting method with respect to physical processes and spatial variables. The original system of integrodifferential equations is split into two subsystems: one of partial differential equations without sources and one of ordinary differential equations (ODE) with sources. The general solution of the ODE system and the energy equation is written in quadratures based on total energy conservation in a cell. A feature of the schemes is that a new approximation is used for the numerical fluxes through the cell interfaces. The fluxes are found along characteristics with the interaction between radiation and matter taken into account. For smooth solutions, the schemes approximating the transfer equations on spatially uniform grids are second-order accurate in time and space. As an example, numerical results for Fleck’s test problems are presented that confirm the increased accuracy and efficiency of the method.

Bibliographie

[1] Zeldovich Ya. B., Raizer Yu. P., Fizika udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavlenii, Nauka, M., 1966

[2] Marchuk G. I., Lebedev V. I., Chislennye metody v teorii perenosa neitronov, Atomizdat, M., 1971 | MR | Zbl

[3] Chetverushkin B. N., Matematicheskoe modelirovanie zadach dinamiki izluchayuschego gaza, Nauka, M., 1985 | Zbl

[4] Yanenko N. N., Metod drobnykh shagov resheniya mnogomernykh zadach matematicheskoi fiziki, Nauka, Novosibirsk, 1967

[5] Karlson B., Bell Dzh., “Reshenie transportnogo uravneniya Sn-metodom”, Fizika yadernykh reaktorov, Atomizdat, M., 1959, 408–432

[6] Karlson B., Bell Dzh., “Chislennoe reshenie zadach kineticheskoi teorii neitronov”, Teoriya yadernykh reaktorov, Gosatomizdat, M., 1963, 243–258

[7] Bass L. P., Voloschenko A. M., Germogenova T. A., Metody diskretnykh ordinat v zadachakh o perenose izlucheniya, IPM im. M. V. Keldysha RAN, M., 1986

[8] Vladimirov V. S., “Chislennoe reshenie uravneniya dlya sfery”, Vychisl. matem., VTs AN SSSR, M., 1958, 3–33

[9] Nikiforova A. V., Tarasov V. A., Troschiev V. E., “O reshenii kineticheskikh uravnenii divergentnym metodom kharakteristik”, Zh. vychisl. matem. i matem. fiz., 12:4 (1972), 1041–1048 | MR | Zbl

[10] Andreev E. S., Kozmanov M. Yu., “Metodika “SVET” dlya resheniya zadach perenosa teplovogo izlucheniya”, VANT. Ser. Metodiki i programmy chisl. resheniya zadach matem. fiz., 1985, no. 2, 50–59

[11] Featrier P. C. R., Acad. Sci. Paris, 258 (1964), 3198–3210

[12] Rybicki G., J. of Quantitative Spectroscopy and Radiative Transfer, 11 (1971), 589–596 | DOI

[13] Goldin V. Ya., “Kvazidiffuzionnyi metod resheniya kineticheskogo uravneniya”, Zh. vychisl. matem. i matem. fiz., 4:6 (1964), 1078–1087 | MR

[14] Anistratov D. Yu., Aristova E. N., Goldin V. Ya., “Nelineinyi metod resheniya zadach perenosa izlucheniya v srede”, Matem. modelirovanie, 8:12 (1996), 3–28 | MR | Zbl

[15] Zuev A. I., “Primenenie metoda Nyutona–Kantorovicha dlya resheniya zadachi o rasprostranenii neravnovesnogo izlucheniya”, Zh. vychisl. matem. i matem. fiz., 13:3 (1973), 792–798 | MR

[16] Gusev V. Yu., Kozmanov M. Yu., Rachilov E. B., “Metod resheniya neyavnykh raznostnykh uravnenii, approksimiruyuschikh sistemy uravnenii perenosa i diffuzii izlucheniya”, Zh. vychisl. matem. i matem. fiz., 24:12 (1984), 1842–1849 | MR | Zbl

[17] Fedotova L. P., Shagaliev R. M., “Konechno-raznostnyi metod KM-metod dlya dvumernykh nestatsionarnykh protsessov perenosa v mnogogruppovom kineticheskom priblizhenii”, Matem. modelirovanie, 3:6 (1991), 29–41

[18] Gadzhiev A. D., Romanova E. M., Seleznev V. N., Shestakov A. A., “Metodika TOM4-KD dlya matematicheskogo modelirovaniya dvumernykh uravnenii perenosa izlucheniya v mnogogruppovom kvazidiffuzionnom priblizhenii”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2001, no. 4, 48–59

[19] Karlykhanov N. G., “Postroenie optimalnykh mnogodiagonalnykh metodov resheniya zadach perenosa izlucheniya”, Zh. vychisl. matem. i matem. fiz., 37:4 (1997), 494–498 | MR | Zbl

[20] Samarskii A. A., Vvedenie v teoriyu raznostnykh skhem, Nauka, M., 1971 | MR | Zbl

[21] Smelov V. V., Lektsii po teorii perenosa neitronov, Atomizdat, M., 1978 | MR

[22] Dolgoleva G. V., “Chislennoe reshenie sistemy uravnenii, opisyvayuschei perenos izlucheniya i vzaimodeistvie ego s veschestvom”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 1991, no. 1, 58–60

[23] Mikhals D., Zvezdnye atmosfery, Mir, M., 1982

[24] Gadzhiev A. D., Shestakov A. A., “Metod vydeleniya diagonalnoi matritsy dlya chislennogo resheniya uravnenii perenosa izlucheniya v R1-priblizhenii po skheme “Romb””, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2006, no. 1

[25] Groshev E. V., “O primenenii metoda Raibiki k protsessu resheniya sistemy uravnenii perenosa izlucheniya iteratsiyami po granichnym usloviyam”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2010, no. 1, 39–47

[26] Gadzhiev A. D., Seleznev V. N., Shestakov A. A., “DS-n metod s iskusstvennoi dissipatsiei i VDM-metod uskoreniya iteratsii dlya chislennogo resheniya dvumernogo uravneniya perenosa teplovogo izlucheniya v kineticheskoi modeli”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2003, no. 4, 33–46

[27] Tikhomirov B. P., “Raznostnaya skhema “KREST” dlya sistemy uravnenii perenosa luchistoi energii”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2009, no. 2, 21–36 | MR

[28] Kakhaner D., Mouler K., Nesh S., Chislennye metody i programmnoe obespechenie, Mir, M., 2001

[29] Bagrinovskii K. A., Godunov S. K., “Raznostnye skhemy dlya mnogomernykh zadach”, Dokl. AN SSSR, 1957, 431–433 | MR

[30] Godunov S. K., Zabrodin A. V., Ivanov M. Ya. i dr., Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976 | MR | Zbl

[31] Yanenko N. N., Yaushev I. K., “Ob odnoi absolyutno ustoichivoi skheme integrirovaniya uravnenii gidrodinamiki”, Raznostnye metody resheniya zadach matematicheskoi fiziki, Tr. MIAN, 74, 1966

[32] Moiseev N. Ya., Silanteva I. Yu., “Raznostnye skhemy proizvolnogo poryadka approksimatsii dlya resheniya lineinykh uravnenii perenosa s postoyannymi koeffitsientami metodom Godunova s antidiffuziei”, Zh. vychisl. matem. i matem. fiz., 48:6 (2008), 1282–1293 | MR

[33] Makotra O. A., Moiseev N. Ya., Silanteva I. Yu. i dr., “Cimmetrichnye raznostnye skhemy pokomponentnogo rasschepleniya i ekvivalentnye im skhemy prediktor-korrektor dlya resheniya mnogomernykh zadach gazovoi dinamiki metodom Godunova”, Zh. vychisl. matem. i matem. fiz., 48:9 (2009), 1659–1672

[34] Moiseev N. Ya., “Neyavnye raznostnye skhemy beguschego scheta povyshennoi tochnosti”, Zh. vychisl. matem. i matem. fiz., 51:5 (2011), 920–935 | MR | Zbl

[35] Fleck J. A., Cummings J. D., “An Implicit Monte Carlo Scheme for Calculating Time and Frequency Dependent Nonlinear Radiation Transport”, J. of Comput. Phys., 8 (1971), 313–342 | DOI | MR | Zbl

[36] Zavyalov V. V., Kozmanov M. Yu., Seleznev V. N. i dr., “Rezultaty chislennykh raschetov odnomernykh testovykh zadach perenosa izlucheniya”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2005, no. 3, 26–36

[37] Reed W. H., “New Difference Schemes for the Neutron Transport Equation”, Nucl. Sci. Eng., 46:2 (1971), 309–314

[38] Barysheva N. M., Zuev A. I., Karlykhanov N. G. i dr., “Neyavnaya skhema dlya chislennogo modelirovaniya fizicheskikh protsessov v lazernoi plazme”, Zh. vychisl. matem. i matem. fiz., 22:2 (1982), 401–410 | MR

[39] Shokin Yu. I., Yanenko H. H., Metod differentsialnogo priblizheniya, Nauka, Novosibirsk, 1979 | MR

[40] Samarskii A. A., Vabischevich P. N., Additivnye skhemy dlya zadach matematicheskoi fiziki, Nauka, M., 2001 | Zbl

[41] Yanenko N. N., Demidov G. V., “Metod slaboi approksimatsii kak konstruktivnyi metod postroeniya resheniya zadachi Koshi”, Nekotorye voprosy vychisl. i prikladnoi matem., Nauka, Novosibirsk, 1966

[42] Fryazinov I. V., “Ekonomichnye simmetrizovannye skhemy resheniya kraevykh zadach dlya mnogomernogo uravneniya parabolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 8:2 (1968), 436–443

[43] Baker G. A., Oliphant T. A., “An implicit numerical method for solving the two-dimensional heat equation”, Quart. Appl. Math., 17:4 (1960) | MR

[44] Godunov S. K., Zabrodin A. V., “O raznostnykh skhemakh vtorogo poryadka tochnosti dlya mnogomernykh zadach”, Zh. vychisl. matem. i matem. fiz., 2:4 (1962), 706–708 | MR | Zbl

[45] Strang G., “On the construction and comparison of difference schemes”, SIAM J. Numer. Anal., 5 (1968), 506–507 | DOI | MR

[46] Moiseev N. Ya., “Monotonnye raznostnye skhemy povyshennoi tochnosti dlya resheniya zadach gazovoi dinamiki metodom Godunova s antidiffuziei”, Zh. vychisl. matem. i matem. fiz., 51:4 (2011), 723–734 | MR | Zbl

[47] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1968 | MR