Geodesic
Application of M-PML absorbing boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part~I: reflectivity
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 4, pp. 333-344.

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This paper presents a detailed study of the construction of reflectionless boundary conditions for anisotropic elastic problems. A Multiaxial Perfectly Matched Layer (M-PML) approach is considered. With a proper stabilization parameter, the M-PML ensures solution stability for arbitrary anisotropic media. It is proved that this M-PML modification is not perfectly matched, and the reflectivity the M-PML exceeds that of the standard PML. Moreover, the reflection coefficient linearly depends on the stabilization parameter. A problem of constructing an optimal stabilization parameter is formulated as follows: find a minimal possible parameter that ensures stability. This problem is considered in a second paper on this work. 
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     title = {Application of {M-PML} absorbing boundary conditions to the numerical simulation of wave propagation in anisotropic media. {Part~I:} reflectivity},
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M. N. Dmitriev; V. V. Lisitsa. Application of M-PML absorbing boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part~I: reflectivity. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 4, pp. 333-344. http://geodesic.mathdoc.fr/item/SJVM_2011_14_4_a0/

[1] Godunov S. K., Romenskii E. I., Elementy mekhaniki sploshnykh sred i zakony sokhraneniya, Izd-vo SO RAN, Novosibirsk, 1998

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

[3] Lisitsa V. V., “Nerasscheplennyi idealno soglasovannyi sloi dlya sistemy uravnenii dinamicheskoi teorii uprugosti”, Sib. zhurn. vychisl. matematiki. RAN. Sib. otd-nie, 10:3 (2007), 285–297

[4] Alpert B., Greengard L., Hagstrom T., “Rapid evaluation of nonreflecting boundary kernels for 15 time-domain wavepropagation”, SAIM J. Numer. Anal., 37 (2000), 1138–1164 | DOI | MR | Zbl

[5] Alpert B., Greengard L., Hagstrom T., “Nonreflecting boundary conditions for the time-dependent wave equation”, J. Comput. Phys., 180 (2002), 270–296 | DOI | MR | Zbl

[6] Appelo D., Hagstrom T., Kreiss G., “Perfectly matched layers for hyperbolic systems: general formulation, well-posednessand stability”, SIAM J. Appl. Math., 67 (2006), 1–23 | DOI | MR | Zbl

[7] Appelo D., Kreiss G., “A new absorbing layer for elastic waves”, J. Comput. Phys., 215 (2005), 642–660 | DOI | MR

[8] Asvadurov S., Druskin V., Guddati M. N., Knizhnerman L., “On optimal finite-difference approximation of PML”, SAIM J. Numer. Anal., 41:1 (2003), 287–305 | DOI | MR | Zbl

[9] Bayliss A., Turkel E., “Radiation boundary conditions for wave-like equations”, Comm. Pure and Appl. Math., 33 (1980), 707–725 | DOI | MR | Zbl

[10] Bécache E., Fauqueux S., Joly P., “Stability of perfectly matched layers, group velocities and anisotropic waves”, J. Comput. Phys., 188:2 (2003), 399–433 | DOI | MR | Zbl

[11] Bécache E., Givoli D., Hagstrom T., “High-order absorbing boundary conditions for anisotropic and convective wave equations”, J. Comput. Phys., 229:4 (2010), 1099–1129 | DOI | MR | Zbl

[12] Berenger J.-P., “A perfectly matched layer for the absorption of electromagnetic waves”, J. Comput. Phys., 114 (1994), 185–200 | DOI | MR | Zbl

[13] Collino F., Monk P. B., “Optimizing the perfectly matched layer”, Comput. Methods Appl. Mech. Eng., 164 (1998), 157–171 | DOI | MR | Zbl

[14] Collino F., Tsogka C., “Application of the perfectly matched layer absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media”, Geophysics, 66 (2001), 294–307 | DOI

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

[16] Hagstrom T., Goodrich J., “Accurate radiation boundary conditions for the linearized euler equations in Cartesian domains”, SIAM J. Sci. Comput., 24 (2003), 770–795 | DOI | MR

[17] Higdon R., “Numerical absorbing boundary conditions for the wave equation”, Math. Comput., 49 (1987), 65–90 | DOI | MR | Zbl

[18] Hiptmair R., Schadle A., “Non-reflecting boundary conditions for Maxwell's equations”, Computing, 71 (2003), 265–292 | DOI | MR | Zbl

[19] Komatitsch D., Martin R., “An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation”, Geophysics, 72:5 (2007), SM155–SM167 | DOI

[20] Lindman E., “Free space boundary conditions for the time dependent wave equation”, J. Comput. Phys., 18 (1975), 66–78 | DOI | Zbl

[21] 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

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

[23] Lisitsa V., “Optimal discretization of PML for elasticity problems”, Electron. Trans. Numer. Anal., 30 (2008), 258–277 | MR | Zbl

[24] Lubich C., Schadle A., “Fast convolution for non-reflecting boundary conditions”, SIAM J. Sci. Comput., 24 (2002), 161–182 | DOI | MR | Zbl

[25] Meza-Fajardo Kristel C., Papageorgiou Apostolos S., “A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: stability analysis”, Bulletin of the Seismological Society of America, 98:4 (2008), 1811–1836 | DOI

[26] Petropoulos P., “Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell's equations in rectangular, cylindrical and spherical coordinates”, SIAM J. Appl. Math., 60 (2000), 1037–1058 | DOI | MR | Zbl

[27] Savadatti S., Guddati M. N., “Absorbing boundary conditions for scalar waves in anisotropic media. I: Time harmonic modeling”, J. Comput. Phys., 229:19 (2010), 6696–6714 | DOI | MR | Zbl

[28] Savadatti S., Guddati M. N., “Absorbing boundary conditions for scalar waves in anisotropic media. II: Time-dependent modeling”, J. Comput. Phys., 229:18 (2010), 6644–6662 | DOI | MR | Zbl

[29] Sofronov I. L., “Artificial boundary conditions of absolute transparency for two and three-dimensional external time-dependent scattering problems”, Euro. J. Appl. Math., 9:6 (1998), 561–588 | DOI | MR | Zbl

[30] Tsynkov S. V., “Numerical solution of problems on unbounded domain. A review”, Applied Numerical Mathematics, 27:4 (1998), 465–532 | DOI | MR | Zbl

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