Bicompact scheme for linear inhomogeneous transport equation in a case of a big optical width

E. N. Aristova

Matematičeskoe modelirovanie, Tome 25 (2013) no. 10, pp. 3-18.

Résumé

It have been considered bicompact Rogov schemes for solving linear inhomogeneous transport equation which is adequate for description of particle transport in media. New approach for energy dependence of distribution function on a base of Lebesque averaging methods leads to broadening of ranges in which absorption coefficient should be vary in comparison with the multigroup approach. New method of solution monotonization has been suggested for numerical solving problems containing regions with big optical widths. This method considerably improve accuracy of bicompact schemes in a case of solution nondifferentiability and tends it to the accuracy of conservative-characteristic schemes.

Détail

Keywords: bicompact schemes, opacity coefficient, optical width, lebesgue averaging method.
Mots-clés : transport equation

Bibliographie

[1] Aristova E. N., Baidin D. F., Rogov B. V., “Bikompaktnye skhemy dlya neodnorodnogo lineinogo uravneniya perenosa”, Matematicheskoe modelirovanie, 25:5 (2013), 55–66

[2] Rogov B. V., Mikhailovskaya M. N., “Bikompaktnye skhemy chetvertogo poryadka approksimatsii dlya giperbolicheskikh uravnenii”, DAN, 430:4 (2010), 470–474 | MR | Zbl

[3] Rogov B. V., Mikhailovskaya M. N., “Monotonnye bikompaktnye skhemy dlya lineinogo uravneniya perenosa”, Matematicheskoe modelirovanie, 23:6 (2011), 98–110 | MR | Zbl

[4] Rogov B. V., Mikhailovskaya M. N., “Monotonnye bikompaktnye skhemy dlya lineinogo uravneniya perenosa”, DAN, 436:5 (2011), 600–605 | MR | Zbl

[5] Rogov B. V., Mikhailovskaya M. N., “Monotonnaya vysokotochnaya kompaktnaya skhema beguschego scheta dlya kvazilineinykh uravnenii giperbolicheskogo tipa”, DAN, 440:2 (2011), 172–177 | MR | Zbl

[6] Mikhailovskaya M. N., Rogov B. V., “Bikompaktnye monotonnye skhemy dlya mnogomernogo lineinogo uravneniya perenosa”, Matematicheskoe modelirovanie, 23:10 (2011), 107–116 | MR | Zbl

[7] Mikhailovskaya M. N., Rogov B. V., “Monotonnye kompaktnye skhemy beguschego scheta dlya sistem uravnenii giperbolicheskogo tipa”, ZhVM i MF, 52:4 (2012), 672–695 | MR

[8] Bakirova M. I., Karpov V. Ya., Mukhina M. I., “Kharakteristiko-interpolyatsionnyi metod resheniya uravneniya perenosa”, Differentsialnye uravneniya, 22:7 (1986), 1141–1148 | MR | Zbl

[9] Aristova E. N., Baidin D. F., Goldin V. Ya., “Dva varianta ekonomichnogo metoda resheniya uravneniya perenosa v $r-z$ geometrii na osnove perekhoda k peremennym Vladimirova”, Matematicheskoe modelirovanie, 18:7 (2006), 43–52 | MR | Zbl

[10] Mathews K. A., “On the propagation of rays in discrete ordinates”, Nucl. Sci. and Eng., 123 (1999), 155–180

[11] Lathrop K. D., “Spatial differencing of the transport equation: Positivity vs. accuracy”, J. Comput. Phys., 4:4 (1969), 475–490 | DOI

[12] Nikolaeva O. V., “Spetsialnye setochnye approksimatsii dlya uravneniya perenosa v silno geterogennykh sredakh s $x-y$ geometriei”, ZhVM i MF, 44:5 (2004), 883–903 | MR | Zbl

[13] Shilkov A. V., Metod lebegovskogo osredneniya uravneniya perenosa chastits v srede, preprint No 100, IPM im. M. V. Keldysha AN SSSR, M., 1987 | MR

[14] Shilkov A. V., Matematicheskaya model dlya opisaniya neravnovesnoi izluchayuschei plazmy, preprint No 125, IPM AN SSSR, M., 1988

[15] Shilkov A. V., Tsvetkova I. L., Shilkova S. V., “Sistema kodov i bank dannykh ATRAD dlya pretsizionnykh raschetov atmosfernoi radiatsii”, Matemat. modelirovanie, 6:7 (1994), 91–102 | MR

[16] Shilkov A. V., Tsvetkova I. L., Shilkova S. V., “Sistema ATRAD dlya raschetov atmosfernoi radiatsii: lebegovskoe osrednenie spektrov i sechenii pogloscheniya”, Matemat. modelirovanie, 9:6 (1997), 3–24

[17] Tsvetkova I. L., Shilkov A. V., “Osrednenie uravneniya perenosa v rezonansno pogloschayuschei srede”, Matemat. modelirovanie, 1:1 (1989), 91–100 | MR | Zbl

[18] Tsvetkova I. L., Shilkov A. V., Realizatsiya metoda lebegovskogo osredneniya uravneniya perenosa izlucheniya, preprint No 100, IPM im. M. V. Keldysha AN SSSR, M., 1989 | MR

[19] Shilkov A. V., “Metody osredneniya sechenii i energeticheskogo spektra v zadachakh perenosa neitronov”, Matem. modelirovanie, 3:2 (1991), 63–81 | MR

[20] Shilkov A. V., “Averaging of cross-sections and energy spectrum in neutron transport problems”, Advances in Math., Comp., and Reactor Physics, Proc. of the Int. Topical Meeting (1991, Pittsburgh, USA), v. 3, 13.1.1

[21] Mozheiko S. V., Tsvetkova I. L., Shilkov A. V., “Raschet perenosa izlucheniya v goryachem vozdukhe”, Matem. modelirovanie, 4:1 (1992), 65–82 | MR

[22] Shilkov A. V., “Generalized Multigroup Approximation and Lebesgue Averaging Method in Particle Transport Problems”, Transp. Theory and Stat. Physics, 23:6 (1994), 781–814 | DOI | MR | Zbl

[23] Shilkov A. V., Shilkova S. V., Goldin V. Ya., Aristova E. N., “Ekonomichnye pretsizionnye raschety atmosfernoi radiatsii na osnove sistemy ATRAD”, DAN, 369:5 (1999), 611–613

[24] Shilkov A. V., Shilkova S. V., Goldin V. Ya., Aristova E. N., “Sistema ATRAD dlya raschetov atmosfernoi radiatsii: raschety solnechnogo izlucheniya dlya letnei atmosfery srednikh shirot”, Matematicheskoe modelirovanie, 11:5 (1999), 117–125

[25] Shilkov A. V., Gertsev M. N., Aristova E. N., Shilkova S. V., “Metodika etalonnykh “line-by-line” raschetov atmosfernoi radiatsii”, Kompyuternye issledovaniya i modelirovanie, 4:3 (2012)

[26] Rosenbrock H., “Some general implicit processes for numerical solution of differential equations”, Computer Journal, 5:4 (1963), 329–330 | DOI | MR | Zbl

[27] Kalitkin N. N., “Poluneyavnye skhemy dlya zadach bolshoi zhestkosti”, v. 1, Entsiklopediya nizkotemperaturnoi plazmy, Seriya B, VII-1, ed. Yu. P. Popov, Yanus-K, M., 153–171

[28] Aristova E. N., Rogov B. V., “O realizatsii granichnykh uslovii v bikompaktnykh skhemakh dlya lineinogo uravneniya perenosa”, Matematicheskoe modelirovanie, 24:10 (2012), 3–14 | Zbl

[29] Dekker K., Verver Ya., Ustoichivost metodov Runge–Kutty dlya zhestkikh nelineinykh differentsialnykh uravnenii, Mir, M., 1988, 334 pp. | MR

[30] Khairer E., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Zhestkie i differentsialno-algebraicheskie zadachi, Mir, M., 1999, 685 pp.