Time splitting for wave equations in random media

Guillaume Bal; Lenya Ryzhik

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 6, page 961-987
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Bal, Guillaume, and Ryzhik, Lenya. "Time splitting for wave equations in random media." ESAIM: Mathematical Modelling and Numerical Analysis 38.6 (2010): 961-987. <http://eudml.org/doc/194251>.

@article{Bal2010,
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. },
author = {Bal, Guillaume, Ryzhik, Lenya},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {High frequency waves in random media; time splitting; multiscale analysis.},
language = {eng},
month = {3},
number = {6},
pages = {961-987},
publisher = {EDP Sciences},
title = {Time splitting for wave equations in random media},
url = {http://eudml.org/doc/194251},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Bal, Guillaume
AU - Ryzhik, Lenya
TI - Time splitting for wave equations in random media
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 6
SP - 961
EP - 987
AB - 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.
LA - eng
KW - High frequency waves in random media; time splitting; multiscale analysis.
UR - http://eudml.org/doc/194251
ER -

References

top
  1. G. Bal, On the self-averaging of wave energy in random media. SIAM Multiscale Model. Simul.2 (2004) 398–420.  Zbl1072.35505
  2. G. Bal and L. Ryzhik, Time reversal for classical waves in random media. C. R. Acad. Sci. Paris I333 (2001) 1041–1046.  Zbl1033.74022
  3. G. Bal and L. Ryzhik, Time reversal and refocusing in random media. SIAM J. Appl. Math.63 (2003) 1475–1498.  Zbl1126.76360
  4. G. Bal, A. Fannjiang, G. Papanicolaou and L. Ryzhik, Radiative transport in a periodic structure. J. Statist. Phys.95 (1999) 479–494.  Zbl0964.82048
  5. G. Bal, G. Papanicolaou and L. Ryzhik, Radiative transport limit for the random Schrödinger equations. Nonlinearity15 (2002) 513–529.  Zbl0999.60061
  6. G. Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation. Stochastics Dynamics4 (2002) 507–531.  Zbl1020.35126
  7. G. Bal, T. Komorowski and L. Ryzhik, Self-averaging of the Wigner transform in random media. Comm. Math. Phys.242 (2003) 81–135.  Zbl1037.35108
  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.  Zbl1006.65112
  9. C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror. Asymptot. Anal.29 (2002) 157–182.  Zbl1015.35005
  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).  
  12. J.F. Clouet and J.-P. Fouque, A time-reversal method for an acoustical pulse propagating in randomly layered media. Wave Motion25 (1997) 361–368.  Zbl0920.73051
  13. G.C. Cohen, Higher-order numerical methods for transient wave equations. Scientific Computation, Springer-Verlag, Berlin (2002).  
  14. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6, Springer-Verlag, Berlin (1993).  
  15. D.R. Durran, Nunerical Methods for Wave equations in Geophysical Fluid Dynamics. Springer, New York (1999).  
  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.  Zbl1028.82010
  17. M. Fink, Time reversed acoustics. Physics Today50 (1997) 34–40.  
  18. M. Fink, Chaos and time-reversed acoustics. Physica Scripta90 (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.  Zbl0881.35099
  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.  Zbl1053.82030
  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.  Zbl0922.65071
  23. A. Ishimaru, Wave Propagation and Scattering in Random Media. New York, Academics (1978).  Zbl0873.65115
  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).  Zbl0848.35068
  25. P.-L. Lions and T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana9 (1993) 553–618.  Zbl0801.35117
  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.  Zbl0928.65109
  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.  Zbl1029.65098
  28. G. Papanicolaou, L. Ryzhik and K. Solna, The parabolic approximation and time reversal. Matem. Contemp.23 (2002) 139–159.  Zbl1027.76049
  29. G. Papanicolaou, L. Ryzhik and K. Solna, Statistical stability in time reversal. SIAM J. App. Math.64 (2004) 1133–1155.  Zbl1065.35058
  30. F. Poupaud and A. Vasseur, Classical and quantum transport in random media. J. Math. Pures Appl.82 (2003) 711–748.  Zbl1035.82037
  31. L. Ryzhik, G. Papanicolaou and J.B. Keller, Transport equations for elastic and other waves in random media. Wave Motion24 (1996) 327–370.  Zbl0954.74533
  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).  
  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.  Zbl0964.82508
  35. G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal.5 (1968) 507–517.  Zbl0184.38503
  36. F. Tappert, The parabolic approximation method, Lect. notes Phys., Vol. 70, Wave propagation and underwater acoustics. Springer-Verlag (1977).  
  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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.