Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T21:41:37.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Time splitting for wave equations in random media

Published online by Cambridge University Press:  15 December 2004

Guillaume Bal  and
Lenya Ryzhik
Guillaume Bal
Affiliation:
Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027, USA. gb2030@columbia.edu.
Lenya Ryzhik
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA. ryzhik@math.uchicago.edu.
Get access

Abstract

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.

Keywords

High frequency waves in random mediatime splittingmultiscale analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 6 , November 2004 , pp. 961 - 987
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bal, G., On the self-averaging of wave energy in random media. SIAM Multiscale Model. Simul. 2 (2004) 398–420. CrossRef
Bal, G. and Ryzhik, L., Time reversal for classical waves in random media. C. R. Acad. Sci. Paris I 333 (2001) 1041–1046. CrossRef
Bal, G. and Ryzhik, L., Time reversal and refocusing in random media. SIAM J. Appl. Math. 63 (2003) 1475–1498. CrossRef
Bal, G., Fannjiang, A., Papanicolaou, G. and Ryzhik, L., Radiative transport in a periodic structure. J. Statist. Phys. 95 (1999) 479–494. CrossRef
Bal, G., Papanicolaou, G. and Ryzhik, L., Radiative transport limit for the random Schrödinger equations. Nonlinearity 15 (2002) 513–529. CrossRef
Bal, G., Papanicolaou, G. and Ryzhik, L., Self-averaging in time reversal for the parabolic wave equation. Stochastics Dynamics 4 (2002) 507–531. CrossRef
Bal, G., Komorowski, T. and Ryzhik, L., Self-averaging of the Wigner transform in random media. Comm. Math. Phys. 242 (2003) 81–135. CrossRef
Bao, W., Jin, S. and Markowich, P.A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comp. Phys. 175 (2002) 487–524. CrossRef
Bardos, C. and Fink, M., Mathematical foundations of the time reversal mirror. Asymptot. Anal. 29 (2002) 157–182.
Blomgren, P., Papanicolaou, G. and Zhao, H., Super-resolution in time-reversal acoustics. J. Acoust. Soc. Am. 111 (2002) 230–248. CrossRef
S. Chandrasekhar, Radiative Transfer. Dover Publications, New York (1960).
Clouet, J.F. and Fouque, J.-P., A time-reversal method for an acoustical pulse propagating in randomly layered media. Wave Motion 25 (1997) 361–368. CrossRef
G.C. Cohen, Higher-order numerical methods for transient wave equations. Scientific Computation, Springer-Verlag, Berlin (2002).
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6, Springer-Verlag, Berlin (1993).
D.R. Durran, Nunerical Methods for Wave equations in Geophysical Fluid Dynamics. Springer, New York (1999).
Erdös, L. and Yau, H.T., Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Comm. Pure Appl. Math. 53 (2000) 667–735. 3.0.CO;2-5>CrossRef
Fink, M., Time reversed acoustics. Physics Today 50 (1997) 34–40. CrossRef
Fink, M., Chaos and time-reversed acoustics. Physica Scripta 90 (2001) 268–277. CrossRef
Gérard, P., Markowich, P.A., Mauser, N.J. and Poupaud, F., Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323–380. 3.0.CO;2-C>CrossRef
Golse, F., Jin, S. and Levermore, C.D., The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method. SIAM J. Numer. Anal. 36 (1999) 1333–1369. CrossRef
Hodgkiss, W., Song, H., Kuperman, W., Akal, T., Ferla, C. and Jackson, D., A long-range and variable focus phase-conjugation experiment in a shallow water. J. Acoust. Soc. Am. 105 (1999) 1597–1604. CrossRef
Hou, T.Y., Wu, X. and Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 227 (1999) 913–943. CrossRef
A. Ishimaru, Wave Propagation and Scattering in Random Media. New York, Academics (1978).
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).
Lions, P.-L. and Paul, T., Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553–618. CrossRef
Markowich, P., Pietra, P. and Pohl, C., Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit. Numer. Math. 81 (1999) 595–630. CrossRef
Markowich, P., Pietra, P., Pohl, C. and Stimming, H.P., Wigner-measure, A analysis of the Dufort-Frankel scheme for the Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 1281–1310. CrossRef
Papanicolaou, G., Ryzhik, L. and Solna, K., The parabolic approximation and time reversal. Matem. Contemp. 23 (2002) 139–159.
Papanicolaou, G., Ryzhik, L. and Solna, K., Statistical stability in time reversal. SIAM J. App. Math. 64 (2004) 1133–1155. CrossRef
Poupaud, F. and Vasseur, A., Classical and quantum transport in random media. J. Math. Pures Appl. 82 (2003) 711–748. CrossRef
Ryzhik, L., Papanicolaou, G. and Keller, J.B., Transport equations for elastic and other waves in random media. Wave Motion 24 (1996) 327–370. CrossRef
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).
P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Academic Press, New York (1995).
Spohn, H., Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17 (1977) 385–412. CrossRef
Strang, G., On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 507–517. CrossRef
F. Tappert, The parabolic approximation method, Lect. notes Phys., Vol. 70, Wave propagation and underwater acoustics. Springer-Verlag (1977).
B.J. Uscinski, The elements of wave propagation in random media. McGraw-Hill, New York (1977).
Uscinski, B.J., Analytical solution of the fourth-moment equation and interpretation as a set of phase screens. J. Opt. Soc. Am. 2 (1985) 2077–2091. CrossRef