ARTICLE

Discontinuous Galerkin method for solving 2D dissipative seismic wave equations

XIJUN HE1 CHUJUN QIU2 JIANQIANG SUN3
Show Less
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: 8 July 2021 | Accepted: 16 January 2022 | Published: 1 April 2022
© 2022 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/ )
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, SanFrancisco.
  2. Ba, J., Xu, W., Fu, L.Y., Carcione, J.M. and Zhang, L., 2017. Rock anelasticity due topatchy saturation and fabric heterogeneity: A double double-porosity model of wavepropagation. J. Geophys. Res.-Solid Earth, 122: 1949-1976.
  3. Cai, W., Zhang, H. and Wang, Y., 2017. Dissipation-preserving spectral element methodfor damped seismic wave equations. J. Computat. Phys., 350: 260-279.
  4. Carcione, J.M., 1993. Seismic modeling in viscoelastic media. Geophysics, 58: 110-120.
  5. Carcione, J.M. and Cavallini, F., 1994. A rheological model for anelastic anisotropicmedia with applications to seismic wave propagation. Geophys. J. Internat., 119:338-348.
  6. Carcione, J.M. and Quiroga-Goode, G., 1995. Some aspects of the physics and numericalmodeling of Biot compressional waves. J. Computat. Acoust., 3: 261-280.
  7. Cockburn, B. and Shu, C.W., 1989. TVB Runge-Kutta local projection discontinuous
  8. Galerkin finite element method for conservation laws: II. General framework.Mathemat. Computat., 52: 411-435.de Basabe, J.D., Sen, M.K. and Wheeler, M.F., 2008. The interior penalty discontinuous
  9. Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Internat.,175: 83-93.de Hoop, A., 1960. A modification of Cagniard’s method for solving seismic pulseproblems. Appl. Sci. Res. Sec., B 8: 349-356.de la Puente, J., Dumbser, M., Kaser, M. and Igel, H., 2008. Discontinuous Galerkinmethods for wave propagation in poroelastic media. Geophysics, 73(5): T77-T97.
  10. Emmerich, H. and Korn, M., 1987. Incorporation of attenuation into time domaincomputations of seismic wave fields. Geophysics, 52: 1252-1264.
  11. Etienne, V., Chaljub, E., Virieux, J. and Glinsky, N., 2010. An hp-adaptivediscontinuous Galerkin finite-element method for 3-D elastic wave modelling.Geophys. J. Internat., 183: 941-962.
  12. Ferroni, A., Antonietti, P. F., Mazzieri, I. and Quarteroni, A., 2017. Dispersion-dissipation analysis of 3-D continuous and discontinuous spectral element methodsfor the elastodynamics equation. Geophys. J. Internat., 211: 1554-1574.
  13. He, X.J., Yang, D.H. and Wu, H., 2015. A weighted Runge-Kutta discontinuous
  14. Galerkin method for wavefield modelling. Geophys. J. Internat., 200: 1389-1410.
  15. Hestholm, S., 1999. Three-dimensional finite-difference viscoelastic wave modellingincluding surface topography. Geophys. J. Internat., 139: 852-878.
  16. Hu, F.Q., Hussaini, M.Y. and Rasitarinera, P., 1999. An analysis of the discontinuous
  17. Galerkin method for wave propagation problems. J. Computat. Phys., 151: 921-946.
  18. Kaser, M. and Dumbser, M., 2006. An arbitrary high-order discontinuous Galerkinmethod for elastic waves on unstructured meshes - I: The two-dimensional isotropiccase with external source terms. Geophys. J. Internat., 166: 855-877.
  19. Kaser, M., Dumbser, M., de La Puente, J. and Igel, H., 2007. An arbitrary high-order
  20. Discontinuous Galerkin method for elastic waves on unstructured meshes-III.
  21. Viscoelastic attenuation. Geophys. J. Internat., 168: 224-242.
  22. Kristek, J. and Moczo, P., 2003. Seismic-wave propagation in viscoelastic media withmaterial discontinuities: a 3D fourth-order staggered-grid finite-difference modeling.Bull. Seismol. Soc. Am., 93: 2273-2280.
  23. Lahivaara, T. and Huttunen, T., 2010. A non-uniform basis order for the discontinuous
  24. Galerkin method of the 3D dissipative wave equation with perfectly matched layer. J.Computat. Phys., 229: 5144-5160.
  25. Lamb, H., 1904. On the propagation of tremors over the surface of an elastic solid. Phil.Trans. Roy. Soc. London., Ser. A., 203: 1-42.
  26. Lambrecht, L., Lamert, A., Friederich, W. MGller, T. and Boxberg, M.S., 2018. A nodaldiscontinuous Galerkin approach to 3-D viscoelastic wave propagation in complexgeological media. Geophys. J. Internat., 212: 1570-1587.
  27. Li, Q.Y., Yi, D.Y. and Wang, N.C., 2008. Numerical Analysis. Tsinghua UniversityPress (in Chinese).
  28. McLachlan, R.I. and Quispel, G.R.W., 2002. Splitting methods. Acta Numer., 11:341-434.
  29. Moczo, P. and Kristek, J., 2005. On the rheological models used for time-domainmethods of seismic wave propagation. Geophys. Res. Lett., 32: L01306.
  30. Moczo, P., Kristek, J. and Halada, L., 2000. 3D fourth-order staggered-gridfinite-difference schemes: Stability and grid dispersion. Bull. Seismol. Soc. Am., 90:587-603.
  31. Niu, B.H. and Sun C.Y., 2007. Half-Space homogeneous isotropic--viscoelastic mediumand seismic wave propagation (in Chinese). Geological Publishing House, Beijing.
  32. Reed, W.H. and Hill, T., 1973. Triangular mesh methods for the neutron transportequation. Los Alamos Scientific Lab., New Mex., U.S.A.
  33. Riviere, B., Shaw, S. and Whiteman, J.R., 2007. Discontinuous Galerkin finite elementmethods for dynamic linear solid viscoelasticity problems. Numerical Methods forPartial Differential Equations, 23:1149-1166.
  34. Robertsson, J.O., Blanch, J.O. and Symes, W.W., 1994. Viscoelastic finite-differencemodeling. Geophysics, 59: 1444-1456.
  35. Sarmany, D., Botchev, M.A. and van der Vegt, J.J., 2007. Dispersion and dissipationerror in high-order Runge-Kutta discontinuous Galerkin discretisations of theMaxwell equations. J. Scient. Comput., 33: 47-74.
  36. Virieux, J., 1986. P-SV wave propagation in heterogeneous media: velocity-stressfinite-difference method. Geophysics, 51: 889-901.
  37. Wang, N., Li, J., Borisov, D., Gharti, H.N., Shen, Y., Zhang, W. and Savage, B., 2019.
  38. Modeling three-dimensional wave propagation in anelastic models with surfacetopography by the optimal strong stability preserving Runge-Kutta method. J.Geophys. Res.: Solid Earth, 124: 890-907.
  39. Wang, N. and Zhou, Y., 2014. A weak dispersion 3D wave field simulation method: Apredictor-corrector method of the Implicit Runge-Kutta Scheme. J. Seismic Explor.,23: 431-462.
  40. Wilcox, L.C., Stadler, G., Burstedde, C. and Ghattas, O., 2010. A high-orderdiscontinuous Galerkin method for wave propagation through coupledelastic—acoustic media. J. Computat. Phys., 229: 9373-9396.
  41. Yang, D., He, X., Ma, X., Zhou, Y. and Li, J., 2016. An optimal nearly analyticdiscrete-weighted Runge-Kutta discontinuous Galerkin hybrid method for acousticwavefield modeling. Geophysics, 81(5): T251-T263.
  42. Yang, H.Z. and Du, Q.Z., 2003. Finite-element methods for viscoelastic and azimuthallyanisotropic media. Acta Phys. Sinica (in Chinese), 52: 2010-2014.
Share
Back to top
Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing