Geodesic
Flux-splitting schemes for parabolic equations with mixed derivatives
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1314-1328 
P. N. Vabishchevich. Flux-splitting schemes for parabolic equations with mixed derivatives. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1314-1328. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a8/
@article{ZVMMF_2013_53_8_a8,
     author = {P. N. Vabishchevich},
     title = {Flux-splitting schemes for parabolic equations with mixed derivatives},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1314--1328},
     year = {2013},
     volume = {53},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a8/}
}
TY  - JOUR
AU  - P. N. Vabishchevich
TI  - Flux-splitting schemes for parabolic equations with mixed derivatives
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2013
SP  - 1314
EP  - 1328
VL  - 53
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a8/
LA  - ru
ID  - ZVMMF_2013_53_8_a8
ER  - 
%0 Journal Article
%A P. N. Vabishchevich
%T Flux-splitting schemes for parabolic equations with mixed derivatives
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2013
%P 1314-1328
%V 53
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a8/
%G ru
%F ZVMMF_2013_53_8_a8

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

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.

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

[2] Knabner P., Angermann L., Numerical methods for elliptic and parabolic partial differential equations, Springer, New York, 2003 | MR

[3] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[4] Fridman A., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968

[5] Protter M. H., Weinberger H. F., Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N.J., 1967 | MR | Zbl

[6] Matus P., Rybak I., “Difference schemes for elliptic equations with mixed derivatives”, Comput. Appl. Math., 4:4 (2004), 494–505 | MR | Zbl

[7] Marchuk G. I., Handbook of Numerical Analysis, Splitting and alternating direction methods, v. 1, Elsevier Sci. Publ. B.V., Amsterdam, 1990 | MR

[8] Samarskii A. A., Vabischevich P. N., Additivnye skhemy dlya zadach matematicheskoi fiziki, Nauka, M., 1999 | MR

[9] Degtyarev L. M., Favorskii A. P., “Potokovyi variant metoda progonki”, Zh. vychisl. matem. i matem. fiz., 8:3 (1968), 679–684 | MR

[10] Degtyarev L. M., Favorskii A. P., “Potokovyi variant metoda progonki dlya raznostnykh zadach s silno menyayuschimisya koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 9:1 (1969), 211–218 | MR | Zbl

[11] Brezzi F., Fortin M., Mixed and Hybrid Finite Element Methods, Springer, New York, 1991 | MR | Zbl

[12] Roberts J. E., Thomas J. M., Handbook of Numerical Analysis, Mixed and hybrid methods, v. II, Elsevier Sci. Publ. B.V., Amsterdam, 1991 | MR

[13] Vabischevich P. N., “Potokovye skhemy rasschepleniya dlya parabolicheskikh zadach”, Zh. vychisl. matem. i matem. fiz., 52:8 (2012), 1415–1425

[14] Samarskii A. A., Gulin A. V., Ustoichivost raznostnykh skhem, Nauka, M., 1973 | Zbl

[15] Samarskii A. A., Matus P. P., Vabishchevich P. N., Difference schemes with operator factors, Kluwer Acad. Pub., Dordrecht–Boston–London, 2002 | MR | Zbl

[16] Konovalov A. N., “Sopryazhenno-faktorizovannye modeli v zadachakh matematicheskoi fiziki”, Sib. zh. vychisl. matem., 1:1 (1998), 25–58 | MR

[17] Samarskii A. A., Andreev V. B., Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976 | MR | Zbl

[18] Horn R. A., Johnson C. R., Matrix analysis, Cambridge University Press, 1990 | MR | Zbl

[19] Douglas Jr. J., Rachford H. H., “On the numerical solution of heat conduction problems in two and three space variables”, Trans. Amer. Math. Soc., 82:2 (1956), 421–439 | DOI | MR | Zbl

[20] Peaceman D. W., Rachford H. H., “The numerical solution of parabolic and elliptic differential equations”, J. SIAM, 3:1 (1955), 28–41 | MR | Zbl

[21] Vabischevich P. N., “Postroenie skhem rasschepleniya na osnove approksimatsii operatora perekhoda”, Zh. vychisl. matem. i matem. fiz., 52:2 (2012), 253–262

[22] McKee S., Mitchell A. R., “Alternating direction methods for parabolic equations in two space dimensions with a mixed derivative”, Comput. J., 13:1 (1970), 81–86 | DOI | MR | Zbl

[23] Hout K. J., Mishra C., “Stability of the modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term”, Math. Comput. Simulat., 81:11 (2011), 2540–2548 | DOI | MR | Zbl

[24] Vabishchevich P. N., “On a new class of additive (splitting) operator-difference schemes”, Math. Comput., 81:277 (2012), 267–276 | DOI | MR | Zbl

[25] Samarskii A. A., “Ob odnom ekonomichnom algoritme chislennogo resheniya sistem differentsialnykh i algebraicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 4:3 (1964), 580–585 | MR

[26] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978 | MR