Geodesic
Bicompact Rogov schemes for the multidimensional inhomogeneous linear transport equation at large optical depths
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 10, pp. 1684-1697 
E. N. Aristova; S. V. Martynenko. Bicompact Rogov schemes for the multidimensional inhomogeneous linear transport equation at large optical depths. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 10, pp. 1684-1697. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_10_a8/
@article{ZVMMF_2013_53_10_a8,
     author = {E. N. Aristova and S. V. Martynenko},
     title = {Bicompact {Rogov} schemes for the multidimensional inhomogeneous linear transport equation at large optical depths},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1684--1697},
     year = {2013},
     volume = {53},
     number = {10},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_10_a8/}
}
TY  - JOUR
AU  - E. N. Aristova
AU  - S. V. Martynenko
TI  - Bicompact Rogov schemes for the multidimensional inhomogeneous linear transport equation at large optical depths
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2013
SP  - 1684
EP  - 1697
VL  - 53
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_10_a8/
LA  - ru
ID  - ZVMMF_2013_53_10_a8
ER  - 
%0 Journal Article
%A E. N. Aristova
%A S. V. Martynenko
%T Bicompact Rogov schemes for the multidimensional inhomogeneous linear transport equation at large optical depths
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2013
%P 1684-1697
%V 53
%N 10
%U http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_10_a8/
%G ru
%F ZVMMF_2013_53_10_a8

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

Bicompact Rogov schemes intended for the numerical solution of the inhomogeneous transport equation are extended to the multidimensional case. A factorized modification of the method without using splitting in directions or introducing additional half-integer spatial points is proposed. As its original counterpart, the scheme is fourth-order accurate in space and third-order accurate in time. In the case of one dimension, a higher order accurate scheme on a minimal stencil is constructed using the node values of the unknown function and, in addition, its integral averages over a spatial cell. In the case of two dimensions, the set of unknowns in a given cell is expanded to four. The resulting system of equations is solved for the expanded set of variables by the running calculation method, which reflects the characteristic properties of the transport equation without explicit use of characteristics. In the case of large optical depths and a piecewise differentiable solution, a monotonization procedure is proposed based on the Rosenbrock scheme with complex coefficients.

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

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

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

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

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

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

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

[8] Rogov B. V., “Vysokotochnaya monotonnaya kompaktnaya skhema beguschego scheta dlya mnogomernykh uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 53:2 (2013), 94–104

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

[10] Aristova E. N., “Bikompaktnye skhemy dlya neodnorodnogo lineinogo uravneniya perenosa v sluchae bolshikh opticheskikh tolschin”, Matem. modelirovanie, 25:10 (2013), 3–16

[11] Shilkov A. V., Metod lebegovskogo osredneniya uravneniya perenosa chastits v srede, Preprint IPM AN SSSR, No 100, M., 1987

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

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

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

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

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

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

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

[19] 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

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

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

[22] 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”, Matem. modelirovanie, 11:5 (1999), 117–125

[23] 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), 553–562

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

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

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

[27] Kalitkin N. N., “Poluneyavnye skhemy dlya zadach bolshoi zhestkosti”, Entsiklopediya nizkotemperaturnoi plazmy. Ser. B, Ch. 1, v. VII-1, Yanus-K, M., 2008, 153–171