Geodesic
On peculiarities of the Lebedev scheme for simulation of elastic wave propagation in anisotropic media
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 2, pp. 155-167 
V. V. Lisitsa; D. M. Vishnevsky. On peculiarities of the Lebedev scheme for simulation of elastic wave propagation in anisotropic media. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 2, pp. 155-167. http://geodesic.mathdoc.fr/item/SJVM_2011_14_2_a3/
@article{SJVM_2011_14_2_a3,
     author = {V. V. Lisitsa and D. M. Vishnevsky},
     title = {On peculiarities of the {Lebedev} scheme for simulation of elastic wave propagation in anisotropic media},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {155--167},
     year = {2011},
     volume = {14},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2011_14_2_a3/}
}
TY  - JOUR
AU  - V. V. Lisitsa
AU  - D. M. Vishnevsky
TI  - On peculiarities of the Lebedev scheme for simulation of elastic wave propagation in anisotropic media
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2011
SP  - 155
EP  - 167
VL  - 14
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SJVM_2011_14_2_a3/
LA  - ru
ID  - SJVM_2011_14_2_a3
ER  - 
%0 Journal Article
%A V. V. Lisitsa
%A D. M. Vishnevsky
%T On peculiarities of the Lebedev scheme for simulation of elastic wave propagation in anisotropic media
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2011
%P 155-167
%V 14
%N 2
%U http://geodesic.mathdoc.fr/item/SJVM_2011_14_2_a3/
%G ru
%F SJVM_2011_14_2_a3

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

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Main attention is being given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the scheme approximates a system of equations which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts”. Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of a problem on a sufficiently smooth solution. The derived results are of importance for the construction of reflectionless boundary conditions, development of a heterogeneous Lebedev scheme, approximation of point sources, etc.

[1] Vavakin A. S., Salganik R. L., “Effektivnye uprugie kharakteristiki tel s izolirovannymi treschinami, polostyami i zhestkimi neodnorodnostyami”, Izvestiya AN SSSR. Mekhanika tverdogo tela, 1978, no. 2, 95–107

[2] Goldin S. V., Seismicheskie volny v anizotropnykh sredakh, Izd-vo SO RAN, Novosibirsk, 2008

[3] Lebedev V. I., “Raznostnye analogi ortogonalnykh razlozhenii, osnovnykh differentsialnykh operatorov dlya nekotorykh kraevykh zadach matematicheskoi fiziki. I”, Zhurn. vychisl. matematiki i mat. fiziki, 4:3 (1964), 449–465 | MR | Zbl

[4] Lisitsa V. V., “Optimalnye setki dlya resheniya volnovogo uravneniya s peremennymi koeffitsientami”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 8:3 (2005), 219–229 | Zbl

[5] Shokin Yu. I., Yanenko N. N., Metod differentsialnogo priblizheniya. Primenenie k gazovoi dinamike, Nauka, Novosibirsk, 1985 | MR | Zbl

[6] Asvadurov S., Druskin V., Moskow S., “Optimal grids for anisotropic problems”, Electron. Trans. Numer. Anal., 26 (2007), 55–81 | MR | Zbl

[7] Backus G. E., “Long-wave elastic anisotropy produced by horizontal layering”, J. of Geophysical Research, 67:11 (1962), 4427–4440 | DOI

[8] Bansal R., Sen M. K., “Finite-difference modelling of S-wave splitting in anisotropic media”, Geophysical Prospecting, 56:3 (2008), 293–312 | DOI

[9] Bernth H., Chapman C., “A comparisson of finite-difference grids for anisotropic elastic modelling”, Expanded Abstracts of 72nd EAGE Conference and Exposition, Spain, Barcelona, 2010, G008

[10] Davydycheva S., Druskin V., Habashy T., “An efficient finite-difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media”, Geophysics, 68:5 (2003), 1525–1535 | DOI

[11] Engquist B., Majda A., “Absorbing boundary conditions for the numerical simulation of waves”, Math. Comp., 31:139 (1977), 629–651 | DOI | MR | Zbl

[12] Igel H., Mora P., Riollet B., “Anisotropic wave propagation through finite-difference grids”, Geophysics, 60:4 (1995), 1203–1216 | DOI

[13] Kachanov M., “Effective elastic properties of cracked solids: Critical review of some basic concepts”, Applied Mechanical Review, 45:8 (1992), 304–335 | DOI

[14] Levander A. R., “Fourth-order finite-difference P-SV seismograms”, Geophysics, 53:11 (1988), 1425–1436 | DOI

[15] Lisitsa V., Lys E., “Reflectionless truncation of target area for axially symmetric anisotropic elasticity”, J. of Computational and Applied Mathematics, 234:6 (2010), 1803–1809 | DOI | MR | Zbl

[16] Lisitsa V., Vishnevskiy D., “Lebedev scheme for the numerical simulation of wave ropagation in 3D anisotropic elasticity”, Geophysical Prospecting, 58:4 (2010), 619–635 | DOI

[17] Pissarenko D., Reshetova G. V., Tcheverda V. A., “3D finite-difference synthetic acoustic logging in cylindrical coordinates”, Geophysical Prospecting, 57:3 (2009), 367–377 | DOI

[18] Saenger E. H., Bohlen T., “Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid”, Geophysics, 69:2 (2004), 583–591 | DOI

[19] Saenger E. H., Gold N., Shapiro S. A., “Modeling the propagation of the elastic waves using a modified finite-difference grid”, Wave Motion, 31:1 (2000), 77–92 | DOI | MR | Zbl

[20] Schoenberg M., Muir F., “A calculus for finely layered anisotropic media”, Geophysics, 54:5 (1989), 581–589 | DOI | MR

[21] Sengupta M., Bachrach R., Bakulin A., “Relationship between velocity and anisotropy perturbations and anomalous stress field around salt bodies”, The Leading Edge, 28:5 (2009), 598–605 | DOI

[22] Thomsen L., “Weak elastic anisotropy”, Geophysics, 51:10 (1986), 1954–1966 | DOI

[23] Virieux J., “P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method”, Geophysics, 51:4 (1986), 889–901 | DOI