Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T07:06:06.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Published online by Cambridge University Press:  28 May 2015

Wensheng Zhang ,
Li Tong  and
Eric T. Chung
Wensheng Zhang*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing P. O. Box 2719, 100190 China
Li Tong*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing P. O. Box 2719, 100190 China
Eric T. Chung*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
*
Corresponding author.Email address:zws@lsec.cc.ac.cn
Corresponding author.Email address:tongli@lsec.cc.ac.cn
Corresponding author.Email address:tschung@math.cuhk.edu.hk
Get access

Abstract

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.

Keywords

35L0565M06765Y05Acoustic wave equationimplicit schemesADILODstability conditiondispersion curveMPI parallel computations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 2 , May 2012 , pp. 205 - 228
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alford, R. M., Kelly, K. R., and Boore, D. M., Accuracy of finite-difference modeling of acoustic wave equations, Geophysics, 39 (1974), pp. 834–842.CrossRefGoogle Scholar
[2]Bayliss, A., Jordan, K. E., Lemesurier, B. J., and Turkel, E., A fourth-order accurate finite-difference scheme for the computation of elastic waves, Bull. Seis. Soc. Am., 76 (1986), pp. 1115–1132.CrossRefGoogle Scholar
[3]Cerjan, C., Kosloff, D., Kosloff, R., and Reshef, M., A nonreflecting boundary condition for discrete acoustic and elastic wave equations, Geophysics, 50 (1985), pp. 705–708.CrossRefGoogle Scholar
[4]Chung, E. T. and Engquist, B., Optimal discontinuous Galerkin methods for wave propagation, SIAM J. Numer. Anal., 44 (2006), pp. 2131–2158.CrossRefGoogle Scholar
[5]Chung, E. T.Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimen sions, SIAM J. Numer. Anal., 47 (2009), pp. 3820–3848.CrossRefGoogle Scholar
[6]Ciarlet, P. G., The finite element method for elliptic problems, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978.Google Scholar
[7]Claerbout, J. F., Imaging the earth’s interior, Blackwell Scientific Publications Inc., 1982.Google Scholar
[8]Cohen, G., Joly, P, Roberts, J. E., and Tordjman, N., Higher order triangular finite elements with mass lumping for the wave equation, SIAM J. Numer. Anal., 38 (2001), pp. 2047–2078.CrossRefGoogle Scholar
[9]Dablain, M. A., The application of higher-order differencing for the scalar wave equation, Geophysics, 51 (1986), pp. 54–66.CrossRefGoogle Scholar
[10]Fairweather, G., and Mitchell, A. R., A high Accuracy alternating direction method for the wave equation, IMAJ. Appl. Math., 1 (1965), pp. 309–316.CrossRefGoogle Scholar
[11]Fornberg, B., On a Fourier methodfor the integration of hyperbolic equations, SIAM J. Numer. Anal., 12 (1975), pp. 509–528.CrossRefGoogle Scholar
[12], Fornberg, B., The pseudospectral method: accurate representation of interfaces in elastic wave calcula tions, Geophysics, 53 (1988), pp. 625–637.CrossRefGoogle Scholar
[13], Fornberg, B., High-order finite differences and the pseudospectral method on staggered grids, SIAM J. Numer. Anal., 27 (1990), pp. 904–918.CrossRefGoogle Scholar
[14]Gazdag, J., Modeling of the acoustic wave equation with transform methods, Geophysics, 46 (1981), pp. 854–859.CrossRefGoogle Scholar
[15]Geiser, J., Higher-order splitting methodfor elastic wave propagation, International Journal of Mathematics and Mathematical Sciences, Volume 2008, Article ID 291968, 31 pages.Google Scholar
[16]Geiser, J., Fourth-order splitting methods for time-dependent differential equations, Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 321–339.Google Scholar
[17]Gourlay, A. R. and Mitchell, A. R., A classification of split difference methods for hyperbolic equations in several space dimensions, SIAM J. Numer. Anal., 6 (1969), pp. 62–71.CrossRefGoogle Scholar
[18]Hua, L. G., Introduction to higher mathematics (in Chinese, Academic Press, Beijing, China, 2009.Google Scholar
[19]Kelly, K. R., Ward, R. W., Treitel, S., and Alford, R. M., Synthetic seismograms: A finite-difference approach, Geophysics, 41 (1976), pp. 2–27.CrossRefGoogle Scholar
[20]Kosloff, D. D. and Baysal, E., Forward modeling by a Fourier method, Geophysics, 47 (1982), pp. 1402–1412.CrossRefGoogle Scholar
[21]Kosloff, D., Kessler, D., Filho, A. Q., Tessmer, E., Behle, A., and Strahilevitz, R., Solution of the equations of dynamic elasticity by a Chebychev spectral method, Geophysics, 55 (1990), pp. 734–748.CrossRefGoogle Scholar
[22]Kosloff, D., Reshef, M., and Lowenthal, D., Elastic wave calculations by the Fourier method, Bull. Seis. Soc. Am., 74 (1984), pp. 875–891.CrossRefGoogle Scholar
[23]Levander, A. R., Fourth-order finite-difference P-SV seismograms, Geophysics, 53 (1988), pp. 1425–1436.CrossRefGoogle Scholar
[24]Marfurt, K. J., Accuracy of finite-difference and finite-element modelling of the scalar and elastic wave euqations, Geophysics, 49 (1984), pp. 533–549.CrossRefGoogle Scholar
[25]Minkoff, S. E., Spatial parallelism of a 3D finite difference velocity-stress elastic wave propagation code, SIAM J. Sci. Comput., 24 (2002), pp. 1–19.CrossRefGoogle Scholar
[26]Orszag, S. A., Spectral methods for problems in complex geometries, J. Comput. Phys., 37 (1980), pp. 70–92.CrossRefGoogle Scholar
[27]Samarskii, A. A., Local one-dimensional difference schemes for multi-dimensional hyperbolic equations in an arbitrary region, Zh. Vychisl. Mat. Mat. Fiz., 4 (964), pp. 638–648.Google Scholar
[28]Sei, A., A family of numerical schemes for the computation of elastic waves, SIAM J. Sci. Comput., 16 (1995), pp. 898–916.CrossRefGoogle Scholar
[29]Thomas, J. W., Numerical partial differential equations: finite difference methods, Springer-verlag New York, Inc., 1995.CrossRefGoogle Scholar
[30]Virieux, J., SH-wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 49 (1984), pp. 1933–1957.CrossRefGoogle Scholar
[31], Virieux, J., P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 51 (1986), pp. 889–901.CrossRefGoogle Scholar
[32]Zhang, G. and Zhang, W., Parallel implementation of2-D prestack depth migration, The Fourth International Conference/Exhibition on High performance Computing in Asia-Pacific Region, Beijing, China, Expanded Abstracts, 2000, pp. 970–975.Google Scholar