AccScience Publishing / JSE / Article
Submit to JSE
Cite this article
84
Views
Journal Browser
Volume | Year
Issue
Search
Forthcoming Issue
Current Issue
View All
News and Announcements
View All
ARTICLE

A high-order weighted Runge-Kutta Discontinuous Galerkin Method for solving 2D acoustic and elastic wave equations in isotropic and anisotropic media

XIJUN HE1,2 ,  XIAORUI YUE2
Show Less
1 Department of Mathematics, School of Mathematics and Statistics, Beijing Technology and Business University (BTBU), Beijing 100048, P.R. China,
2 Department of Mathematics, College of Science, Hainan University, Haikou 570228, P.R. China,
JSE 2019, 28(4), 363–390;
Submitted: 19 November 2018 | Accepted: 10 June 2019 | Published: 1 August 2019
© 2019 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

He, X.J. and Yue, X.R., 2019. A high-order weighted Runge-Kutta Discontinuous Galerkin Method for solving 2D acoustic and elastic wave equations in isotropic and anisotropic media. Journal of Seismic Exploration, 28: 363-391. A high-order weighted Runge-Kutta Discontinuous Galerkin Method for solving 2D acoustic and elastic wave equations in isotropic and anisotropic media is proposed in this paper, which is an extension of the existing first-order and second-order methods to higher-order cases. For this method, second-order seismic wave equations are first transformed into a first-order hyperbolic system, then local Lax-Friedrichs (LLF) numerical flux discontinuous Galerkin formulations for spatial discretization are employed, directly leading to a semi discrete ordinary differential equation (ODE) system. For time discretization, an implicit diagonal Runge-Kutta method is introduced. To avoid solving a large-scale system of linear equations, a two-step explicit iterative process is implemented. In addition, a weighting factor is introduced for the iteration to enrich the method. The basis functions we use are 1 ~ 5' order polynomials, leading to 2'°- and 6 order of spatial accuracy. Numerical properties of the high-order weighted Runge-Kutta Discontinuous Galerkin Method are investigated in detail, including numerical error, stability criteria and numerical dispersion, which validate the superiority of the high order method. The proposed method is then applied to several 2D wave propagation problems in isotropic and anisotropic media, including acoustic-elastic interface problems. Results illustrate that this method can effectively suppress numerical dispersion and provide accurate information on the wave field on coarse mesh. We also compare the proposed method with the finite difference method to investigate the computational efficiency.

Keywords
seismic wave equation
high-order
Discontinuous Galerkin Method
DGM
weighted
numerical dispersion
References
  1. Aki, K. and Richards, P.G., 2002. Quantitative Seismology. W.H. Freeman. & Co.,Sausalito.
  2. Ainsworth, M., Monk, P. and Muniz, W., 2006. Dispersive and dissipative properties ofdiscontinuous Galerkin finite element methods for the second-order waveequation. J. Scientif. Comput., 27: 5-40.
  3. Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., 2002. Unified analysis ofdiscontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Analys.,39: 1749-1779.
  4. Chaljub, E., Moczo, P., Tsuno, S., Bard, P.-Y., Kristek, J., Kaser, M., Stupazzini, M. and
  5. Kristekova, M., 2010. Quantitative comparison of four numerical predictions of3D ground motion in the Grenoble Valley, France. Bull. Seismol. Soc. Am., 100:1427-1455.
  6. Chen, J. and Liu, Q.H., 2013. Discontinuous Galerkin time-domain methods formultiscale electromagnetic simulations: A review. Proc. IEEE, 101: 242-254.
  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.
  9. Cockburn, B. and Shu, C.-W., 2001. Runge-Kutta discontinuous Galerkin methods forconvection-dominated problems. J. Scient. Comput., 16: 173-261.
  10. Dablain, M.A., 1986. The application of high-order differencing to the scalar waveequation. Geophysics, 51: 54-66.
  11. De Basabe, J.D. and Sen, M.K., 2010. Stability of the high-order finite elements foracoustic or elastic wave propagation with high-order time stepping. Geophys. J.Internat., 181: 577-590.
  12. De Basabe, J.D., Sen, M.K. and Wheeler, M.F., 2008. The interior penalty discontinuous
  13. Galerkin method for elastic wave propagation: grid dispersion. Geophys. J.Internat., 175: 83-93.de la Puente, J., Ampuero, J.P. and Kaser, M., 2009. Dynamic rupture modeling onunstructured meshes using a discontinuous Galerkin method. J. Geophys. Res.,Solid Earth: 114.
  14. Dolejsi, V., Holik, M. and Hozman, J., 2011. Efficient solution strategy for thesemi-implicit discontinuous Galerkin discretization of the Navier-Stokesequations. J. Comput. Phys., 230: 4176-4200.
  15. Dumbser, M. and Munz, C.-D., 2005. ADER discontinuous Galerkin schemes foraeroacoustics. Compt. Rend. Mécan., 333: 683-687.
  16. Etienne, V., Chaljub, E., Virieux, J. and Glinsky, N., 2010. An hp-adaptive discontinuous
  17. Galerkin finite-element method for 3-D elastic wave modelling. Geophys. J.Internat., 183: 941-962.
  18. Hairer, E., Norsett, S.P. and Wanner, G., 2006. Solving ordinary differentialequations:nonstiff problems. Springer-Verlag.
  19. He, X. , Yang, D. and Zhou, Y., 2014. A weighted Runge-Kutta discontinuous Galerkinmethod for wavefield modelling. Expanded Abstr., 84th Ann. Internat. SEG Mtg.,Denver.
  20. He, X., Yang, D. and Wu, H., 2015. A weighted Runge-Kutta discontinuous Galerkinmethod for wavefield modelling. Geophys. J. Internat., 200: 1389-1410.
  21. He, X., Yang, D., Ma, X. and Lang, C., 2019. Dispersion-dissipation analysis of thetriangle-based discontinuous Galerkin method for scalar wave equation. Geophys.J. Internat., 218: 1174-1198.
  22. Hesthaven, J.S. and Warburton, T., 2007. Nodal Discontinuous Galerkin Methods:
  23. Algorithms, Analysis, and Applications. Springer Science and Business Media,New York.
  24. Hu, F.Q., Hussaini, M. and Rasetarinera, P., 1999. An analysis of the discontinuous
  25. Galerkin method for wave propagation problems. J. Computat. Phys., 151:921-946.
  26. Kaser, M. and Dumbser, M., 2006. An arbitrary high-order discontinuous Galerkinmethod for elastic waves on unstructured meshes - I. The two-dimensionalisotropic case with external source terms. Geophys. J. Internat., 166: 855-877.
  27. Kaser, M. and Dumbser, M., 2008. A highly accurate discontinuous Galerkin method forcomplex interfaces between solids and moving fluids. Geophysics, 73: T23-T35.
  28. Komatitsch, D. and Tromp, J., 1999. Introduction to the spectral element method forthree-dimensional seismic wave propagation. Geophys. J. Internat., 139: 806-822.
  29. Lambrecht, L., Lamert, A., Friederich, W., Moller, T. and Boxberg, M., 2017. A nodaldiscontinuous Galerkin approach to 3-D viscoelastic wave propagation in complexgeological media. Geophys. J. Internat., 212: 1570-1587.
  30. LeVeque, R.J., 2002. Finite volume methods for hyperbolic problems. CambridgeUniversity Press, Cambridge.
  31. Marfurt, K.J., 1984. Accuracy of finite-difference and finite-element modeling of thescalar and elastic wave equations. Geophysics, 49: 533-549.
  32. Minisini, S., Zhebel, E., Kononov, A. and Mulder, W.A., 2013. Local time stepping withthe discontinuous Galerkin method for wave propagation in 3D heterogeneousmedia. Geophysics, 78: T67-T77.
  33. 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.
  34. Pelties, C., Puente, J., Ampuero, J.P., Brietzke, G.B. and Kaser, M., 2012. Three -dimensional dynamic rupture simulation with a high - order discontinuous
  35. Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res.: SolidEarth, 117.
  36. Reed, W.H. and Hill, T., 1973. Triangular mesh methods for the neutron transportequation. Los Alamos Scientific Lab., NM, U.S.A.
  37. Renac, F., Marmignon, C. and Coquel, F., 2012. Time implicit high-order discontinuousgalerkin method with reduced evaluation cost. SIAM J. Scient. Comput., 34:A370-A394.
  38. Riviere, B. and Wheeler, M.F., 2003. Discontinuous finite element methods for acousticand elastic wave problems. Contemp. Mathemat., 329: 271-282.
  39. Segawa, H., Luo, H. and Nourgaliev, R., 2011. A Diagonally Implicit Runge-Kutta
  40. Method for the Discontinuous Galerkin Solutions of the Navier-Stokes Equations.49th AIAA Aerospace Sciences Mtg. including the New Horizons Forum andAerospace Expos., 685.
  41. Tong, P., Chen, C.-W., Komatitsch, D., Basini, P. and Liu, Q., 2014. High-resolutionseismic array imaging based on an SEM-FK hybrid method. Geophys. J. Internat.,197: 369-395.
  42. Virieux, J., 1986. P-SV wave propagation in heterogeneous media: velocity-stressfinite-difference method. Geophysics, 51: 1933-1942.
  43. Wang, N. and Zhou, Y., 2014. A weak dispersion 3D wave field simulation method: Apredictor-corrector method of the implicit Runge-Kutta scheme. J. SeismicExplor., 23: 431-462.
  44. Yang, D., Peng, J., Lu, M. and Terlaky, T., 2006. Optimal nearly analytic discreteapproximation to the scalar wave equation. Bull. Seismol. Soc. Am., 96:1114-1130.
  45. Yang, D., Wang, N. and Liu, E., 2012. A strong stability-preserving predictor-correctormethod for the simulation of elastic wave propagation in anisotropic media.Commun. Computat. Phys., 12: 1006-1032.
  46. Zhang, J., 1997. Quadrangle-grid velocity-stress finite-difference method forelastic-wave-propagation simulation. Geophys. J. Internat., 131: 127-134.
Share
Back to top
Journal of Seismic Exploration, 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