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Discontinuous Galerkin method for solving 2D dissipative seismic wave equations

XIJUN HE1 ,  CHUJUN QIU2 ,  JIANQIANG SUN3
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1 School of Mathematics and Statistics, Beijing Technology and Business University (BTBU), Beijing 100048, P.R. China.,
2 Department of Mathematical Sciences, Tsinghua University, Beijing 100000, P.R. China.,
3 Department of Mathematical Sciences, Hainan University, Haikou 100000, P.R. China.,
JSE 2022, 31(2), 153–176;
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/ )
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Abstract

He, X.J., Qui, C.J. and Sun, J.Q., 2022. Discontinuous Galerkin method for solving 2D dissipative seismic wave equations. Journal of Seismic Exploration, 31: 153-176. Seismic dissipation widely exists in underground media. To develop a detailed understanding of wave propagation in dissipative media, in this study, we introduce a discontinuous Galerkin (DG) method for solving acoustic and elastic wave equations in D’Alembert media. This method uses the numerical flux-based DG formulations with the explicit 3rd-order total variation diminishing (TVD) time discretization. We first derive an empirical formula for numerical stability conditions, which shows that the relative error of the Courant-Friedrichs-Lewy (CFL) condition numbers between the actual the numerical cases does not exceed 3%. The analyses also show that both the dispersion and dissipation in D’Alembert media are frequency dependent, and have a strong correlation with the dissipation factor. Finally, we present some numerical experiments. The quantitative comparisons of the attenuation ratios of the waveforms show that they are close to the theoretical ones, verifying the findings of the analyses. In particular, for elastic waves, the relative errors between the numerical attenuation ratios and the theoretical ones do not exceed 4%. The simulation of dissipative elastic wave propagation in a model with surface topography indicates our method is capable of dealing with complex geometry.

Keywords
discontinuous Galerkin method
D’Alembert media
dispersion
dissipation
numerical modelling
References
  1. Aki, K. and Richards, P.G., 2002. Quantitative seismology. W.H. Freeman & Co, San
  2. Francisco.
  3. Ba, J., Xu, W., Fu, L.Y., Carcione, J.M. and Zhang, L., 2017. Rock anelasticity due to
  4. patchy saturation and fabric heterogeneity: A double double-porosity model of wave
  5. propagation. J. Geophys. Res.-Solid Earth, 122: 1949-1976.
  6. Cai, W., Zhang, H. and Wang, Y., 2017. Dissipation-preserving spectral element method
  7. for damped seismic wave equations. J. Computat. Phys., 350: 260-279.
  8. Carcione, J.M., 1993. Seismic modeling in viscoelastic media. Geophysics, 58: 110-120.
  9. Carcione, J.M. and Cavallini, F., 1994. A rheological model for anelastic anisotropic
  10. media with applications to seismic wave propagation. Geophys. J. Internat., 119:
  11. 338-348.
  12. Carcione, J.M. and Quiroga-Goode, G., 1995. Some aspects of the physics and numerical
  13. modeling of Biot compressional waves. J. Computat. Acoust., 3: 261-280.
  14. Cockburn, B. and Shu, C.W., 1989. TVB Runge-Kutta local projection discontinuous
  15. Galerkin finite element method for conservation laws: II. General framework.
  16. Mathemat. Computat., 52: 411-435.
  17. de Basabe, J.D., Sen, M.K. and Wheeler, M.F., 2008. The interior penalty discontinuous
  18. Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Internat.,
  19. 175: 83-93.
  20. de Hoop, A., 1960. A modification of Cagniard’s method for solving seismic pulse
  21. problems. Appl. Sci. Res. Sec., B 8: 349-356.
  22. de la Puente, J., Dumbser, M., Kaser, M. and Igel, H., 2008. Discontinuous Galerkin
  23. methods for wave propagation in poroelastic media. Geophysics, 73(5): T77-T97.
  24. Emmerich, H. and Korn, M., 1987. Incorporation of attenuation into time domain
  25. computations of seismic wave fields. Geophysics, 52: 1252-1264.
  26. Etienne, V., Chaljub, E., Virieux, J. and Glinsky, N., 2010. An hp-adaptive
  27. discontinuous Galerkin finite-element method for 3-D elastic wave modelling.
  28. Geophys. J. Internat., 183: 941-962.
  29. Ferroni, A., Antonietti, P. F., Mazzieri, I. and Quarteroni, A., 2017. Dispersion-
  30. dissipation analysis of 3-D continuous and discontinuous spectral element methods
  31. for the elastodynamics equation. Geophys. J. Internat., 211: 1554-1574.
  32. He, X.J., Yang, D.H. and Wu, H., 2015. A weighted Runge-Kutta discontinuous
  33. Galerkin method for wavefield modelling. Geophys. J. Internat., 200: 1389-1410.
  34. Hestholm, S., 1999. Three-dimensional finite-difference viscoelastic wave modelling
  35. including surface topography. Geophys. J. Internat., 139: 852-878.
  36. Hu, F.Q., Hussaini, M.Y. and Rasitarinera, P., 1999. An analysis of the discontinuous
  37. Galerkin method for wave propagation problems. J. Computat. Phys., 151: 921-946.
  38. Kaser, M. and Dumbser, M., 2006. An arbitrary high-order discontinuous Galerkin
  39. method for elastic waves on unstructured meshes - I: The two-dimensional isotropic
  40. case with external source terms. Geophys. J. Internat., 166: 855-877.
  41. Kaser, M., Dumbser, M., de La Puente, J. and Igel, H., 2007. An arbitrary high-order
  42. Discontinuous Galerkin method for elastic waves on unstructured meshes-III.
  43. Viscoelastic attenuation. Geophys. J. Internat., 168: 224-242.
  44. Kristek, J. and Moczo, P., 2003. Seismic-wave propagation in viscoelastic media with
  45. material discontinuities: a 3D fourth-order staggered-grid finite-difference modeling.
  46. Bull. Seismol. Soc. Am., 93: 2273-2280.
  47. Lahivaara, T. and Huttunen, T., 2010. A non-uniform basis order for the discontinuous
  48. Galerkin method of the 3D dissipative wave equation with perfectly matched layer. J.
  49. Computat. Phys., 229: 5144-5160.
  50. Lamb, H., 1904. On the propagation of tremors over the surface of an elastic solid. Phil.
  51. Trans. Roy. Soc. London., Ser. A., 203: 1-42.
  52. Lambrecht, L., Lamert, A., Friederich, W. MGller, T. and Boxberg, M.S., 2018. A nodal
  53. discontinuous Galerkin approach to 3-D viscoelastic wave propagation in complex
  54. geological media. Geophys. J. Internat., 212: 1570-1587.
  55. Li, Q.Y., Yi, D.Y. and Wang, N.C., 2008. Numerical Analysis. Tsinghua University
  56. Press (in Chinese).
  57. McLachlan, R.I. and Quispel, G.R.W., 2002. Splitting methods. Acta Numer., 11:
  58. 341-434.
  59. Moczo, P. and Kristek, J., 2005. On the rheological models used for time-domain
  60. methods of seismic wave propagation. Geophys. Res. Lett., 32: L01306.
  61. Moczo, P., Kristek, J. and Halada, L., 2000. 3D fourth-order staggered-grid
  62. finite-difference schemes: Stability and grid dispersion. Bull. Seismol. Soc. Am., 90:
  63. 587-603.
  64. Niu, B.H. and Sun C.Y., 2007. Half-Space homogeneous isotropic--viscoelastic medium
  65. and seismic wave propagation (in Chinese). Geological Publishing House, Beijing.
  66. Reed, W.H. and Hill, T., 1973. Triangular mesh methods for the neutron transport
  67. equation. Los Alamos Scientific Lab., New Mex., U.S.A.
  68. Riviere, B., Shaw, S. and Whiteman, J.R., 2007. Discontinuous Galerkin finite element
  69. methods for dynamic linear solid viscoelasticity problems. Numerical Methods for
  70. Partial Differential Equations, 23:1149-1166.
  71. Robertsson, J.O., Blanch, J.O. and Symes, W.W., 1994. Viscoelastic finite-difference
  72. modeling. Geophysics, 59: 1444-1456.
  73. Sarmany, D., Botchev, M.A. and van der Vegt, J.J., 2007. Dispersion and dissipation
  74. error in high-order Runge-Kutta discontinuous Galerkin discretisations of the
  75. Maxwell equations. J. Scient. Comput., 33: 47-74.
  76. Virieux, J., 1986. P-SV wave propagation in heterogeneous media: velocity-stress
  77. finite-difference method. Geophysics, 51: 889-901.
  78. Wang, N., Li, J., Borisov, D., Gharti, H.N., Shen, Y., Zhang, W. and Savage, B., 2019.
  79. Modeling three-dimensional wave propagation in anelastic models with surface
  80. topography by the optimal strong stability preserving Runge-Kutta method. J.
  81. Geophys. Res.: Solid Earth, 124: 890-907.
  82. Wang, N. and Zhou, Y., 2014. A weak dispersion 3D wave field simulation method: A
  83. predictor-corrector method of the Implicit Runge-Kutta Scheme. J. Seismic Explor.,
  84. 23: 431-462.
  85. Wilcox, L.C., Stadler, G., Burstedde, C. and Ghattas, O., 2010. A high-order
  86. discontinuous Galerkin method for wave propagation through coupled
  87. elastic—acoustic media. J. Computat. Phys., 229: 9373-9396.
  88. Yang, D., He, X., Ma, X., Zhou, Y. and Li, J., 2016. An optimal nearly analytic
  89. discrete-weighted Runge-Kutta discontinuous Galerkin hybrid method for acoustic
  90. wavefield modeling. Geophysics, 81(5): T251-T263.
  91. Yang, H.Z. and Du, Q.Z., 2003. Finite-element methods for viscoelastic and azimuthally
  92. anisotropic media. Acta Phys. Sinica (in Chinese), 52: 2010-2014.
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Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
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