Time splitting for wave equations in random media
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 6, p. 961-987

Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified regime of the parabolic wave equation in a random medium. The mathematical theory of the convergence and statistical properties of the algorithm is based on the analysis of the Wigner transforms in random media. Our results provide a step toward understanding time and space discretizations that are needed in order for the numerical algorithm to capture the correct macroscopic statistics of the wave energy density in a random medium.

DOI : https://doi.org/10.1051/m2an:2004046
Classification:  65C50,  65M12,  74J20
Keywords: high frequency waves in random media, time splitting, multiscale analysis
@article{M2AN_2004__38_6_961_0,
     author = {Bal, Guillaume and Ryzhik, Lenya},
     title = {Time splitting for wave equations in random media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {6},
     year = {2004},
     pages = {961-987},
     doi = {10.1051/m2an:2004046},
     zbl = {1130.74393},
     mrnumber = {2108940},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_6_961_0}
}
Bal, Guillaume; Ryzhik, Lenya. Time splitting for wave equations in random media. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 6, pp. 961-987. doi : 10.1051/m2an:2004046. http://www.numdam.org/item/M2AN_2004__38_6_961_0/

[1] G. Bal, On the self-averaging of wave energy in random media. SIAM Multiscale Model. Simul. 2 (2004) 398-420. | Zbl 1072.35505

[2] G. Bal and L. Ryzhik, Time reversal for classical waves in random media. C. R. Acad. Sci. Paris I 333 (2001) 1041-1046. | Zbl 1033.74022

[3] G. Bal and L. Ryzhik, Time reversal and refocusing in random media. SIAM J. Appl. Math. 63 (2003) 1475-1498. | Zbl 1126.76360

[4] G. Bal, A. Fannjiang, G. Papanicolaou and L. Ryzhik, Radiative transport in a periodic structure. J. Statist. Phys. 95 (1999) 479-494. | Zbl 0964.82048

[5] G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schrödinger equations. Nonlinearity 15 (2002) 513-529. | Zbl 0999.60061

[6] G. Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation. Stochastics Dynamics 4 (2002) 507-531. | Zbl 1020.35126

[7] G. Bal, T. Komorowski and L. Ryzhik, Self-averaging of the Wigner transform in random media. Comm. Math. Phys. 242 (2003) 81-135. | Zbl 1037.35108

[8] W. Bao, S. Jin and P.A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comp. Phys. 175 (2002) 487-524. | Zbl 1006.65112

[9] C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror. Asymptot. Anal. 29 (2002) 157-182. | Zbl 1015.35005

[10] P. Blomgren, G. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics. J. Acoust. Soc. Am. 111 (2002) 230-248.

[11] S. Chandrasekhar, Radiative Transfer. Dover Publications, New York (1960). | MR 111583

[12] J.F. Clouet and J.-P. Fouque, A time-reversal method for an acoustical pulse propagating in randomly layered media. Wave Motion 25 (1997) 361-368. | Zbl 0920.73051

[13] G.C. Cohen, Higher-order numerical methods for transient wave equations. Scientific Computation, Springer-Verlag, Berlin (2002). | MR 1870851 | Zbl 0985.65096

[14] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6, Springer-Verlag, Berlin (1993). | MR 1295030 | Zbl 0802.35001

[15] D.R. Durran, Nunerical Methods for Wave equations in Geophysical Fluid Dynamics. Springer, New York (1999). | MR 1660086 | Zbl 0918.76001

[16] L. Erdös and H.T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Comm. Pure Appl. Math. 53 (2000) 667-735. | Zbl 1028.82010

[17] M. Fink, Time reversed acoustics. Physics Today 50 (1997) 34-40.

[18] M. Fink, Chaos and time-reversed acoustics. Physica Scripta 90 (2001) 268-277.

[19] P. Gérard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323-380. | Zbl 0881.35099

[20] F. Golse, S. Jin and C.D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method. SIAM J. Numer. Anal. 36 (1999) 1333-1369. | Zbl 1053.82030

[21] W. Hodgkiss, H. Song, W. Kuperman, T. Akal, C. Ferla and D. Jackson, A long-range and variable focus phase-conjugation experiment in a shallow water. J. Acoust. Soc. Am. 105 (1999) 1597-1604.

[22] T.Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 227 (1999) 913-943. | Zbl 0922.65071

[23] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York, Academics (1978). | Zbl 0873.65115

[24] J.B. Keller and R. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations, in Surveys in applied mathematics, J.B. Keller, D. McLaughlin and G. Papanicolaou Eds., Plenum Press, New York (1995). | Zbl 0848.35068

[25] P.-L. Lions and T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553-618. | Zbl 0801.35117

[26] P. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit. Numer. Math. 81 (1999) 595-630. | Zbl 0928.65109

[27] P. Markowich, P. Pietra, C. Pohl and H.P. Stimming, A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 1281-1310. | Zbl 1029.65098

[28] G. Papanicolaou, L. Ryzhik and K. Solna, The parabolic approximation and time reversal. Matem. Contemp. 23 (2002) 139-159. | Zbl 1027.76049

[29] G. Papanicolaou, L. Ryzhik and K. Solna, Statistical stability in time reversal. SIAM J. App. Math. 64 (2004) 1133-1155. | Zbl 1065.35058

[30] F. Poupaud and A. Vasseur, Classical and quantum transport in random media. J. Math. Pures Appl. 82 (2003) 711-748. | Zbl 1035.82037

[31] L. Ryzhik, G. Papanicolaou and J.B. Keller, Transport equations for elastic and other waves in random media. Wave Motion 24 (1996) 327-370. | Zbl 0954.74533

[32] H. Sato and M.C. Fehler, Seismic wave propagation and scattering in the heterogeneous earth. AIP series in modern acoustics and signal processing, AIP Press, Springer, New York (1998). | MR 1488700 | Zbl 0894.73001

[33] P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Academic Press, New York (1995).

[34] H. Spohn, Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17 (1977) 385-412. | Zbl 0964.82508

[35] G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 507-517. | Zbl 0184.38503

[36] F. Tappert, The parabolic approximation method, Lect. notes Phys., Vol. 70, Wave propagation and underwater acoustics. Springer-Verlag (1977). | MR 475274

[37] B.J. Uscinski, The elements of wave propagation in random media. McGraw-Hill, New York (1977).

[38] B.J. Uscinski, Analytical solution of the fourth-moment equation and interpretation as a set of phase screens. J. Opt. Soc. Am. 2 (1985) 2077-2091.