AccScience Publishing / JSE / Article
Submit to JSE
Cite this article
1
Download
5
Views
Journal Browser
Volume | Year
Issue
Search
Forthcoming Issue
Current Issue
View All
News and Announcements
View All
ARTICLE

A fourth-order Runge-Kutta method with eighth-order accuracy and low numerical dispersion for solving the seismic wave equation

CHAOYUAN ZHANG1 ,  LI CHEN2
Show Less
1 College of Mathematics and computer, Dali University, Dali 671003, P.R. China. zcy_km@163.com,
2 College of Engineering, Dali University, Dali 671003, P.R. China.,
JSE 2016, 25(3), 229–255;
Submitted: 9 June 2025 | Revised: 9 June 2025 | Accepted: 9 June 2025 | Published: 9 June 2025
© 2025 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/ )
Download PDF
Cite
XML
Abstract

Zhang, C. and Chen, L., 2016. A fourth-order Runge-Kutta method with eighth-order accuracy and low numerical dispersion for solving the seismic wave equation. Journal of Seismic Exploration, 25: 229-255. In this paper, we give a fourth-order Runge-Kutta method with the eighth-order accuracy and low numerical dispersion for solving the seismic wave equation, which is called the ENAD-FRK method in brief. We first give the theoretical deduction and stability conditions for this new method in detail. And, we derive numerical dispersion relations of the ENAD-FRK method in 2D acoustic case and compare numerical dispersions against the eighth-order Lax-Wendroff correction (LWC) scheme and the eighth-order Staggered-grid (SG) finite difference method. Meanwhile, we compare the memory requirement and the computational efficiency of the proposed method against the eighth-order LWC scheme for modeling 2D seismic wave fields in a two-layer heterogeneous acoustic medium. Last, we apply the ENAD-FRK method to simulate 2D seismic wave propagating in a three-layer homogenous transversely isotropic elastic medium, a two-layer homogenous isotropic elastic medium and a Marmousi model. Simulation results indicate that the ENAD-FRK method can greatly save both computational costs and storage space as contrasted to the eighth-order LWC scheme. Meanwhile, Both comparisons of numerical dispersion analysis and numerical experimental results show that the ENAD-FRK method can effectively suppress numerical dispersion caused by discretizing the seismic wave equation when too coarse grids are used against the eighth-order LWC scheme and the eighth-order SG method.

Keywords
Runge-Kutta method
NAD operator
seismic wave equation
numerical dispersion
wave simulation
References
  1. Blanch, J.O. and Robertsson, A., 1997. A modified Lax-Wendroff correction for wave propagation
  2. in media described by Zener elements. Geophys. J. Internat., 131: 381-386.
  3. Chen, S., Yang, D.H and Deng, X.Y., 2010. An improved algorithm of the fourth-order
  4. Runge-Kutta method and seismic wave-field simulation. Chinese J. Geophys. (in Chinese),
  5. 3: 1196-1206.
  6. Dablain, M.A., 1986. The application of high-order differencing to scalar wave equation.
  7. Geophysics, 51: 54-66.
  8. Dong, L.G., Ma, Z.T., Cao, J.Z., Wang, H.Z., Gong, J.H., Lei, B. and Xu, S.Y., 2000. A
  9. staggered-grid high-order difference method of one-order elastic wave equation. Chinese J.
  10. Geophys. (in Chinese), 43: 411-419.
  11. Kelly, K.R., Wave, R.W. and Treitel, S., 1976. Synthetic seismograms: a finite-difference
  12. approach. Geophysics, 41: 2-27.
  13. Lax, P.D, and Wendroff, B., 1964. Difference schemes for hyperbolic equations with high order
  14. of accuracy. Commun. Pure Appl. Mathem., 17: 381-398.
  15. Moczo, P., Kristek, J. and Halada, L., 2000. 3D 4th-order staggered-grid finite-difference schemes:
  16. stability and grid dispersion. Bull. Seismol. Soc. Am., 90: 587-603.
  17. Moczo, P., Kristek, J., Vavrycuk, V., Archuleta, R.J. and Halada, L., 2002. 3D heterogeneous
  18. staggered-grid finite-difference modeling of seismic motion with volume harmonic and
  19. arithmetic averaging of elastic moduli and densities. Bull. Seismol. Soc. Am., 92:
  20. 3042-3066.
  21. Saenger, E.H., Gold, N. and Shapiro, $.A., 2000. Modeling the propagation of elastic waves using
  22. a modified finite-difference grid. Wave Motion, 31: 77-92.
  23. Tong, P., Yang, D.H., Hua, B.L. and Wang, M.X., 2013. A high-order stereo-modeling method
  24. for solving wave equations. Bull. Seismol. Soc. Am., 103: 811-833.
  25. Vichnevetsky, R., 1979. Stability charts in the numerical approximation of partial differential
  26. equations. a review. Mathemat. Comput. Simul., 21: 170-177.
  27. Virieux, J., 1986. P-SV wave propagation in heterogeneous median: Velocity-stress finite-difference
  28. method. Geophysics, 51: 889-901.
  29. Wang, L., Yang, D.H. and Deng, X.Y., 2009. A WNAD method for seismic stress-field modeling
  30. in heterogeneous media. Chinese J. Geophys. (in Chinese), 52: 1526-1535.
  31. Yang, D.H., Chen, S. and Li, J.Z., 2007a. A Runge-Kutta method using high-order interpolation
  32. approximation for solving 2D acoustic and elastic wave equations. J. Seismic Explor., 16:
  33. 331-353.
  34. Yang, D.H., Liu, E., Zhang, Z.J. and Teng, J.W., 2002. Finite-difference modeling in
  35. two-dimensional anisotropic media using a flux-corrected transport technique. Geophys. J.
  36. Internat., 148: 320-328.
  37. Yang, D.H., Peng, J.M., Lu, M. and Terlaky, T., 2006. Optimal nearly-analytic discrete
  38. approximation to the scalar wave equation. Bull. Seismol. Soc. Am., 96: 1114-1130.
  39. Yang, D.H., Song, G.J., Chen, S. and Hou, B.Y., 2007b. An improved nearly analytical discrete
  40. method: an efficient tool to simulate the seismic response of 2-D porous structures. J.
  41. Geophys. Engin., 4: 40-52.
  42. Yang, D.H., Teng, J.W., Zhang, Z.J. and Liu, E., 2003. A nearly-analytic discrete method for
  43. acoustic and elastic wave equations in anisotropic media. Bull. Seismol. Soc. Am., 93:
  44. 882-890.
  45. Yang, D.H., Tong, P. and Deng, X.Y., 2012. A central difference method with low numerical
  46. dispersion for solving the scalar wave equation. Geophys. Prosp., 60: 885-905.
  47. Yang, D.H., Wang, N., Chen, S. and Song, G.J., 2009. An explicit method based on the implicit
  48. Runge-Kutta algorithm for solving the wave equations. Bull. Seismol. Soc. Am., 99:
  49. 3340-3354.
  50. Zeng, Y.Q. and Liu, Q.H., 2001. A staggered-grid finite-difference method with perfectly matched
  51. layers for poroelastic wave equations. J. Acoust. Soc. Am., 109: 571-2580.
  52. 248 ZHANG & CHEN
  53. Zhang, C.Y., Li, X., Ma, X. and Song, G.J., 2014a. A Runge-Kutta method with using
  54. eighth-order nearly-analytic spatial discretization operator for solving a 2D acoustic wave
  55. equation. J. Seismic Explor., 23: 279-302.
  56. Zhang, C.Y., Ma, X., Yang, L. and Song, G.J., 2014b. Symplectic partitioned Runge-Kutta method
  57. based on the eighth-order nearly analytic discrete operator and its wavefield simulations.
  58. Appl. Geophys., 11: 89-106.
  59. Zhang, Z.J., Wang, G.J. and Harris, J.M., 1999. Multi-component wave-field simulation in viscous
  60. extensively dilatancy anisotropic media. Phys. Earth Planet. Inter., 114: 25-38.
  61. Zheng, H.S,, Zhang, Z.J. and Liu, E., 2006. Non-linear seismic wave propagation in anisotropic
  62. media using the flux-corrected transport technique. Geophys. J. Internat., 65: 943-956.
Share
Back to top
Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept