Geodesic
Finding nonoscillatory solutions to difference schemes for the advection equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 9, pp. 1685-1697 
S. L. Kivva. Finding nonoscillatory solutions to difference schemes for the advection equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 9, pp. 1685-1697. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a12/
@article{ZVMMF_2008_48_9_a12,
     author = {S. L. Kivva},
     title = {Finding nonoscillatory solutions to difference schemes for the advection equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1685--1697},
     year = {2008},
     volume = {48},
     number = {9},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a12/}
}
TY  - JOUR
AU  - S. L. Kivva
TI  - Finding nonoscillatory solutions to difference schemes for the advection equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 1685
EP  - 1697
VL  - 48
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a12/
LA  - ru
ID  - ZVMMF_2008_48_9_a12
ER  - 
%0 Journal Article
%A S. L. Kivva
%T Finding nonoscillatory solutions to difference schemes for the advection equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 1685-1697
%V 48
%N 9
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a12/
%G ru
%F ZVMMF_2008_48_9_a12

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

The advection equation is solved using a weighted adaptive scheme that combines a monotone scheme with the central-difference approximation of the first spatial derivative. The determination of antidiffusion fluxes is treated as an optimization problem. The solvability of the optimization problem is analyzed, and the differential properties of the cost functional are examined. It is shown that the determination of antidiffusion fluxes is reduced to a linear programming problem in the case of an explicit scheme and to a nonlinear programming problem or a sequence of linear programming problems in the case of an implicit scheme. A simplified monotonization algorithm is proposed. Numerical results are presented.

[1] Oran E., Boris Dzh., Chislennoe modelirovanie reagiruyuschikh potokov, Mir, M., 1990 | MR

[2] Zalesak S. T., “Fully multidimensional flux-corrected transport algorithms for fluids”, J. Comput. Phys., 31:3 (1979), 335–362 | DOI | MR | Zbl

[3] Harten A., “On a class of high resolution total-variation-stable finite-difference schemes”, SIAM J. Numer. Analys., 21:1 (1984), 1–23 | DOI | MR | Zbl

[4] Pinchukov V. I., Shu Ch.-V., Chislennye metody vysokikh poryadkov dlya zadach aerogidrodinamiki, Izd-vo SO RAN, Novosibirsk, 2000

[5] Godunov S. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matem. sb., 47 (1959), 271–306 | MR | Zbl

[6] Godunov S. K., Ryabenkii I. S., Raznostnye skhemy, Nauka, M., 1977

[7] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[8] Samarskii A.Ȧ., Gulin A. B., Ustoichivost raznostnykh skhem, Nauka, M., 1973 | Zbl

[9] Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L., Metody splain-funktsii, Nauka, M., 1980 | MR

[10] Ortega Dzh., Reinboldt V., Iteratsionnye metody resheniya nelineinykh sistem uravnenii so mnogimi neizvestnymi, Mir, M., 1975 | MR

[11] Vasilev F. P., Chislennye metody resheniya ekstremalnykh zadach, Nauka, M., 1980 | MR