A truncated implicit high-order finite-difference scheme combined with boundary conditions
Chang Suo-Liang1,2,3 and Liu Yang1
1. State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 100083, China.
2. Taiyuan University of Technology, Taiyuan 030024, China.
3. Shanxi Shandi Geophy-Tech Co. Ltd, Jinzhong, 030600, China.
Abstract：
In this paper, first we calculate finite-difference coefficients of implicit finite-difference methods (IFDM) for the first- and second-order derivatives on normal grids and first-order derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirror-image symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition (ABC) to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.
CHANG Suo-Liang,LIU Yang. A truncated implicit high-order finite-difference scheme combined with boundary conditions[J]. APPLIED GEOPHYSICS, 2013, 10(1): 53-62.
[1]
Bohlen, T., and Saenger, E. H., 2006, Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves: Geophysics, 71, T109 - T115.
[2]
Dablain, M. A., 1986, The application of high-order differencing to the scalar wave equation: Geophysics, 51, 54 - 66.
[3]
Dong, L., Ma, Z., Cao, J., Wang, H., Geng, J., Lei, B., and Xu, S., 2000, A staggered-grid high-order difference method of one-order elastic wave equation: Chinese Journal of Geophysics, 43, 411 - 419.
[4]
Du, Q. Z., Li, B., and Hou B., 2009, Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme: Applied Geophysics, 6(1), 42 - 49.
[5]
Emerman, S., Schmidt, W., and Stephen, R., 1982, An implicit finite-difference formulation of the elastic wave equation: Geophysics, 47, 1521 - 1526.
[6]
Fornberg, B., 1987, The pseudospectral method - comparisons with finite differences for the elastic wave equation: Geophysics, 52, 483 - 501.
[7]
Gold, N., Shapiro, S. A., and Burr, E., 1997, Modeling of high contrasts in elastic media using a modified finite difference scheme: 68th Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 1850 - 1853.
[8]
Graves, R., 1996, Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences: Bulletin of the Seismological Society of America, 86, 1091 - 1106.
[9]
Lele, S. K., 1992, Compact finite difference schemes with spectral-like resolution: Journal of Computational Physics, 103, 16 - 42.
[10]
Liu, Y., Li, C., and Mou, Y., 1998, Finite-difference numerical modeling of any even-order accuracy: Oil Geophysical Prospecting (in Chinese), 33, 1 - 10.
[11]
Liu, Y., and Wei, X. C., 2008, Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media: Applied Geophysics, 5, 107 - 114.
[12]
Liu, Y., and Sen, M. K., 2010a, A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation: Geophysics, 75(2), A1 - A6.
[13]
Liu, Y., and Sen, M. K., 2010b. A hybrid absorbing boundary condition for elastic wave modeling with staggered-grid finite difference: 80th Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 2945 - 2949.
[14]
Liu, Y., and Sen, M. K., 2009a, A practical implicit finite-difference method: Examples from seismic modeling: Journal of Geophysics and Engineering, 6(3), 231 - 249.
[15]
Liu, Y., and Sen, M. K., 2009b, An implicit staggered-grid finite-difference method for seismic modeling: Geophysical Journal International, 179(1), 459 - 474.
[16]
Liu, Y., and Sen, M. K., 2009c, Numerical modeling of wave equation by a truncated high-order finite-difference method: Earthquake Science, 22(2), 205 - 213.
[17]
Moczo, P., Kristek, J., and Halada, L., 2000, 3D fourth-order staggered-grid finite-difference schemes: stability and grid dispersion: Bulletin of the Seismological Society of America, 90, 587 - 603
[18]
Pei, Z., 2004, Numerical modeling using staggered-grid high order finite-difference of elastic wave equation on arbitrary relief surface: Oil Geophysical Prospecting (in Chinese), 39, 629 - 634.
[19]
Robertsson, J., Blanch, J., and Symes, W., 1994, Viscoelastic finite-difference modeling: Geophysics, 59, 1444 - 1456.
[20]
Saenger, E., and Bohlen, T., 2004, Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid: Geophysics, 69, 583 - 591.
2013, Vol. 10 
