Abstract
Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral formulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.
Article PDF
Similar content being viewed by others
References
Bansal R and Sen M K (2008). Finite-difference modelling of S-wave splitting in anisotropic media. Geophysical Prospecting56: 293–312.
Bohlen T and Saenger E H (2006). Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves. Geophysics71: T109–T115.
Chen J B (2007). High-order time discretizations in seismic modeling. Geophysics72: SM115–SM122.
Claerbout J F (1985). Imaging the Earth’s Interior. Blackwell Scientific Publications Inc, Palo. Alto, CA, 90–143.
Crase E (1990). High-order (space and time) finite-difference modeling of the elastic wave equation. 60th Annual International Meeting, SEG, San Francisco, USA, Sept. 23–27, Expanded Abstracts, 987–991.
Dablain M A (1986). The application of high-order differencing to the scalar wave equation. Geophysics51: 54–66.
Dong L G, Ma Z T, Cao J Z, Wang H Z, Geng J H, Lei B and Xu S Y (2000). A staggered-grid high-order difference method of one-order elastic wave equation. Chinese J Geophys43: 411–419.
Emerman S, Schmidt W and Stephen R (1982). An implicit finite-difference formulation of the elastic wave equation. Geophysics47: 1 521–1 526.
Etgen J T and O’Brien M J (2007). Computational methods for large-scale 3D acoustic finite-difference modeling: A tutorial. Geophysics72: SM223–SM230.
Fei T and Liner C L (2008). Hybrid Fourier finite difference 3D depth migration for anisotropic media. Geophysics73: S27–S34.
Fornberg B (1987). The pseudospectral method—Comparisons with finite differences for the elastic wave equation. Geophysics52: 483–501.
Gold N, Shapiro S A and Burr E (1997). Modeling of high contrasts in elastic media using a modified finite difference scheme. 67th Annual International Meeting, SEG, Dallas, USA, Nov. 2–7, Expanded Abstracts, 1 850–1 853.
Graves R (1996). Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bull Seism Soc Amer86: 1 091–1 106.
Hayashi K and Burns D R (1999). Variable grid finite-difference modeling including surface topography. 69th Annual International Meeting, SEG, Houston, USA, Oct. 31–Nov. 5, Expanded Abstracts, 523–527.
Igel H, Mora P and Riollet B (1995). Anistotropic wave propagation through finite-difference grids. Geophysics60: 1 203–1 216.
Kelly K R, Ward R, Treitel W S and Alford R M (1976). Synthetic seismograms: A finite-difference approach. Geophysics41: 2–27.
Kindelan M, Kamel A and Sguazzero P (1990). On the construction and efficiency of staggered numerical differentiators for the wave equation. Geophysics55: 107–110.
Larner K and Beasley C (1987). Cascaded migrations-improving the accuracy of finite-difference migration. Geophysics52: 618–643.
Levander A (1988). Fourth-order finite-difference P-SV seismograms. Geophysics53: 1 425–1 436.
Li Z (1991). Compensating finite-difference errors in 3-D migration and modeling. Geophysics56: 1 650–1 660.
Liu Y, Li C C and Mou Y G (1998). Finite-difference numerical modeling of any even-order accuracy. Oil Geophysical Prospecting33: 1–10 (in Chinese with English abstract).
Liu Y and Wei X C (2008). Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media. Applied Geophysics5: 107–114.
Opršal I and Zahradník J (1999). Elastic finite-difference method for irregular grids. Geophysics64: 240–250.
Pei Z L (2004). Numerical modeling using staggered-grid high order finitedifference of elastic wave equation on arbitrary relief surface. Oil Geophysical Prospecting39: 629–634 (in Chinese with English abstract).
Ristow D and Ruhl T (1994). Fourier finite-difference migration. Geophysics59: 1 882–1 893.
Robertsson J, Blanch J and Symes W (1994). Viscoelastic finite-difference modeling. Geophysics59: 1 444–1 456.
Saenger E and Bohlen T (2004). Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid. Geophysics69: 583–591.
Tessmer E (2000). Seismic finite-difference modeling with spatially varying time steps. Geophysics65: 1 290–1 293.
Virieux J (1986). P-SV wave propagation in heterogeneous media: Velocity stress finite difference method. Geophysics51: 889–901.
Wang Y and Schuster G T (1996). Finite-difference variable grid scheme for acoustic and elastic wave equation modeling. 66th Annual International Meeting, SEG, Denver, USA, Oct. 6–9, Expanded Abstracts, 674–677.
Zhang G, Zhang Y and Zhou H (2000). Helical finite-difference schemes for 3-D. depth migration. 70th Annual International Meeting, SEG, Calgary, Canada, Aug. 6–11, Expanded Abstracts, 862–865.
Zhang H and Zhang Y (2007). Implicit splitting finite difference scheme for multi-dimensional wave simulation. 77th Annual International Meeting, SEG, San Antonio, USA, Sept. 23–28, Expanded Abstracts, 2 011–2 014.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Liu, Y., Sen, M.K. Numerical modeling of wave equation by a truncated high-order finite-difference method. Earthq Sci 22, 205–213 (2009). https://doi.org/10.1007/s11589-009-0205-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11589-009-0205-0