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A Runge-Kutta method with using eighth-order nearly-analytic spatial discretization operator for solving a 2D acoustic wave equation

CHAOYUAN ZHANG1,2 ,  XIAO LI2 ,  XIAO MA2 ,  GUOJIE SONG3
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College of Mathematics and Computer, Dali University, Dali 671003, P.R. China. zcy_km@163.com,
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China.,
Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P.R. China.,
JSE 2014, 23(3), 279–302;
Submitted: 12 March 2013 | Accepted: 6 May 2014 | Published: 1 July 2014
© 2014 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Zhang, C., Li, X., Ma, X. and Song, G., 2014. A Runge-Kutta method with using eighth-order nearly-analytic spatial discretization operator for solving a 2D acoustic wave equation. Journal of Seismic Exploration, 23: 279-302. In this paper, we develop an eighth-order NAD-RK method for solving a 2D acoustic wave equation. The new method uses an eighth-order nearly-analytic discretization (NAD) operator to approximate the high-order spatial derivatives in the wave equation. The wavefield displacements and their gradients are used simultaneously. And we apply a third-order Runge-Kutta (RK) method to solve the semi-discrete ordinary differential equations (ODEs) with respect to time. Thus this method has third-order accuracy in time and can achieve eighth-order accuracy in space. Theoretical properties including stability and errors are analyzed for the eighth-order NAD-RK method in detail. Meanwhile, the numerical dispersion relationship for this method is investigated and the numerical dispersion is tested in our study. The study shows that the eighth-order NAD-RK method has the smallest numerical dispersion and the weakest numerical dispersion anisotropy compared with the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG) method. The computational efficiency of the eighth-order NAD-RK method is also tested. The results show that the eighth-order NAD-RK method needs less computational time and requires less memory than the eighth-order LWC and SG methods. Finally, the eighth-order NAD-RK method is used to simulate acoustic wavefields for two heterogeneous layered models. The simulation results further demonstrate that the eighth-order NAD-RK method can provide high-order accuracy for the complex heterogeneous models and is effective to suppress the numerical dispersion caused by discretizing the acoustic wave equation when too coarse grids are used or strong discontinuities exist in the medium. Thus, the eighth-order NAD-RK method can be potentially used in seismic tomography and large-scale wave propagation problems.

Keywords
RK method
numerical dispersion
acoustic wave equation
NAD operator
References
  1. Blanch, J.O. and Robertsson, A., 1997. A modified Lax-Wendroff correction for wave propagationin media described by Zener elements. Geophys. J. Internat., 131: 381-386.
  2. Chen, J.B., 2007. High-order time discretizations in seismic modeling. Geophysics, 72: 115-122.
  3. Chen, K.H., 1984. Propagating numerical model of elastic wave in anisotropic in homogeneousmedia-finite element method. Expanded Abstr., 54th Ann. Internat. SEG Mtg., Atlanta:631-632.
  4. Chen, S., Yang, D.H. and Deng, X.Y., 2010. An improved algorithm of the fourth-order
  5. Runge-Kutta method and seismic wave-field simulation. Chin. J. Geophys. (in Chinese), 53:1196-1206.
  6. Chen, X.F., 1993. A systematic and efficient method of computing normal modes for multi-layeredhalf space. Geophys. J. Internat., 115: 391-409.
  7. Dablain, M.A., 1986. The application of high-order differencing to scalar wave equation.Geophysics, 51: 54-66.de Hoop, A.T., 1960. A modification of Cagniard’s method for solving seismic pulse problems.Appl. Sci. Res., B8: 349-356.
  8. Dong, L.G., Ma, Z.T., Cao, J.Z., Wang, H.Z., Geng, J.H., Lei, B. and Xu, S.Y., 2000. Astaggered-grid high-order difference method of one-order elastic wave equation. Chin. J.Geophys. (in Chinese), 43: 411-419,
  9. Finkelstein, B. and Kastner, R., 2007. Finite difference time domain dispersion reduction schemes.J. Comput. Phys, 221: 422-438.
  10. Finkelstein, B. and Kastner, R., 2008. A comprehensive new methodology for formulating FDTDschemes with controlled order of accuracy and dispersion. IEEE, Trans. Antenna Propagat,56: 3516-3525.
  11. Huang, B.S., 1992. A program for two-dimensional seismic wave propagation by thepseudo-spectrum method. Comput. Geosci., 18: 289-307.
  12. Kelly, K.R. and Marfurt, K.J. (Eds.), 1990. Numerical Modeling of Seismic Wave Propagation.Geophys. Repr. Ser., 13.
  13. Kelly, K.R. and Wave, R.W., 1976. Synthetic seismograms: a finite-difference approach.Geophysics, 41: 2-27.
  14. Komatitsch, D. and Vilotte, J.P., 1998. The spectral element method: an efficient tool to simulatethe seismic responses of 2D and 3D geological structures. Bull. Seismol. Soc. Am., 88:368-392.
  15. Kosloff, D. and Baysal, E., 1982. Forward modeling by the Fourier method. Geophysics, 47:1402-1412.
  16. Kosloff, D., Reshef, M. and Loewenthal, D., 1984. Elastic wave calculations by the Fouriermethod. Bull. Seismol. Soc. Am., 74: 875-891.
  17. Lax, P.D. and Wendroff, B., 1964. Difference schemes for hyperbolic equations with high orderof accuracy. Communic. Pure Appl. Mathemat., 17: 381-398.
  18. Lele, S.K., 1992. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys.,103: 16-42.
  19. Liu, Y. and Sen, M.K., 2009. A new time-space domain high-order finite-difference method for theacoustic wave equation. J. Comput. Phys., 228: 8779-8806.298 ZHANG, LI, MA & SONG
  20. Marfurt, K.J., 1984. Accuracy of finite-difference and finite-element modeling of the scalar andelastic wave equations. Geophysics, 49: 533-549.
  21. Ma, S. and Liu, P., 2006. Modeling of the perfectly matched layer absorbing boundaries andintrinsic attenuation in explicit finite-element methods. Bull. Seismol. Soc. Am., 96:1779-1794.
  22. Moczo, P., Kristek, J. and Halada, L., 2000. 3D 4th-order staggered-grid finite-difference schemes:stability and grid dispersion. Bull. Seismol. Soc. Am, 90: 587-603.
  23. Moczo, P., Kristek, J., Galis, M., Pazak, P. and Balazovjech, M., 2007. The finite-difference andfinite-element modeling of seismic wave propagation and earthquake motion. Acta Phys.Slovaca, 57: 177-406.
  24. Moczo, P., Kristek, J., Vavrycuk, V., Archuleta, R.J. and Halada, L., 2002. 3D heterogeneousstaggered-grid finite-difference modeling of seismic motion with volume harmonic andarithmetic averaging of elastic moduli and densities. Bull. Seismol. Soc. Am., 92:3042-3066.
  25. Qiu, J.X., Li, T.G. and Khoo, B.C., 2008. Simulations of compressible two-medium flow by
  26. Runge-Kutta discontinuous Galerkin methods with the ghost fluid method. Communic.Comput. Phys., 3: 479-504.
  27. Saenger, E.H., Gold, N. and Shapiro, S.A., 2000. Modeling the propagation of elastic waves usinga modified finite-difference grid. Wave Motion, 31: 77-92.
  28. Seriani, G. and Priolo, E., 1994. Spectral element method for acoustic wave simulation inheterogeneous media. Finite Elem. Analysis Design, 16: 337-348.
  29. Smith, W.D., 1975. The application of finite element analysis to body wave propagation problems.Geophys. J., 42: 747-768.
  30. Tong, P., Yang, D.H., Hua, B.L. and Wang, M.X., 2013. An high-order stereo-modeling methodfor solving wave equations. Bull. Seismol. Soc. Am., 103: 811-833.
  31. Virieux, J., 1986. P-SV wave propagation in heterogeneous median: Velocity-stress finite-differencemethod. Geophysics, 51: 889-901.
  32. Wang, L., Yang, D.H. and Deng, X.Y., 2009. A WNAD method for seismic stress-field modelingin heterogeneous media. Chin. J. Geophys. (in Chinese), 52: 1526-1535.
  33. Yang, D.H., 2002. Finite element method of the elastic wave equation and wave field simulationin two-phase anisotropic media. Chin. J. Geophys. (in Chinese), 45: 575-583.
  34. Yang, D.H., Liu, E., Zhang, Z.J. and Teng, J.W., 2002. Finite-difference modeling intwo-dimensional anisotropic media using a flux-corrected transport technique. Geophys. J.Internat., 148: 320-328.
  35. Yang, D.H., Peng, J.M., Lu, M. and Terlaky, T., 2006. Optimal nearly-analytic discreteapproximation to the scalar wave equation. Bull. Seismol. Soc. Am., 96: 1114-1130.
  36. Yang, D.H., Song, G.J. and Lu, M., 2007. Optimally accurate nearly analytic discrete forwave-field simulation in 3D anisotropic media. Bull. Seismol. Soc. Am., 97, 1557-1569.
  37. Yang, D.H., Song, G.J., Chen, S. and Hou, B.Y., 2007. An improved nearly analytical discretemethod: an efficient tool to simulate the seismic response of 2-D porous structures. J.Geophys. Engin., 4: 40-52.
  38. Yang, D.H., Teng, J.W., Zhang, Z.J. and Liu, E., 2003. A nearly-analytic discrete method foracoustic and elastic wave equations in anisotropic media. Bull. Seismol. Soc. Am., 93:882-890.
  39. Yang, D.H., Tong, P. and Deng, X.Y., 2012. A central difference method with low numericaldispersion for solving the scalar wave equation. Geophys. Prosp., 60, 885-905.
  40. Yang, D.H., Wang, L. and Deng, X.Y., 2010. An explicit split-step algorithm of the implicit
  41. Adams method for solving 2D acoustic and elastic wave equations. Geophys. J. Internat.,180: 291-310.
  42. Yang, D.H., Wang, N., Chen, S. and Song, G.J., 2009. An explicit method based on the implicit
  43. Runge-Kutta algorithm for solving the wave equations. Bull. Seismol. Soc. Am., 99:3340-3354.
  44. Zeng, Y.Q. and Liu, Q.H., 2001. A staggered-grid finite-difference method with perfectly matchedlayers for poroelastic wave equations. J. Acoust. Soc. Am., 109: 2571-2580.A RUNGE-KUTTA METHOD 299
  45. Zhang, Z.J., Wang, G.J. and Harris, J.M., 1999. Multi-component wave-field simulation in viscousextensively dilatancy anisotropic media. Phys. Earth Planet. Inter., 114: 25-38.
  46. Zheng, H.S., Zhang, Z.J. and Liu, E., 2006. Non-linear seismic wave propagation in anisotropicmedia using the flux-corrected transport technique. Geophys. J. Internat., 165: 943-956.
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Journal of Seismic Exploration, Print ISSN: 0963-0651, Published by AccScience Publishing
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