Abstract
We present in this paper a comparison of the dispersion properties for several finite-difference approximations of the acoustic wave equation. We investigate the compact and staggered schemes of fourth order accuracy in space and of second order or fourth order accuracy in time. We derive the computational cost of the simulation implied by a precision criterion on the numerical simulation (maximum allowed error in phase or group velocity). We conclude that for moderate accuracy the staggered scheme of second order in time is more efficient, whereas for very precise simulation the compact scheme of fourth order in time is a better choice. The comparison increasingly favors the lower order staggered scheme as the dimension increases. In three dimensional simulation, the cost of extremely precise simulation with any of the schemes is very large, whereas for simulation of moderate precision the staggered scheme is the least expensive.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alford, R. M., Kelly, K. R., and Boore, D. M. (1974). Accuracy of finite difference modeling of the acoustic wave equation.Geophysics, Vol. 39.
Brown, D. (1984). A Note on the numerical solution of wave equation with piecewise smooth coefficients.Mathematics of Computation, Vol. 42, No. 166.
Cohen, G., and Joly, P. (1986). Fourth order schemes for the 2-D wave equation in homogeneous medium.Proc. of the 56th SEG Ann. Int'l. Meeting and Exposition, Houston.
Cohen, G., and Joly, P. (1987). Description and analysis of schemes, fourth order in time and space, for 1-D and 2-D acoustic equations.Proc. of the 57th SEG Ann. Int'l. Meeting and Exposition, New Orleans.
Cohen, G., and Joly, P. (1990). Fourth order schemes for the heterogeneous acoustic equation.Computer Methods in Applied Mechanics and Engineering.
Dablain, M. (1986). High order differencing for the scalar wave equation.Geophysics, Vol. 51.
Fornberg, B. (1985). The pseudospectral method: comparison with the finite differences for elastic wave equation.Geophysics, Vol. 52, No. 4.
Gazdag, J. (1981). Modeling of the acoustic wave equation with transform methods.Geophysics, Vol. 46, No. 6.
Kern, M., and Symes, W. W. (1993). Loop level parallelisation of a seismic inversion code.Proc. of the 63rd SEG Annual Meeting, Washington D.C.
Kosloff, D. D., and Baysal, E. (1982). Forward modeling by Fourier method.Geophysics, Vol. 47, No. 10.
Lax, P. D., and Wendroff, B. (1964). Difference schemes for hyperbolic equations with high order of accuracy.Comm. on Pure and Appl. Math., Vol. XVII.
Luo, Y., and Schuster, G. (1990). A parsimonous staggered grid differencing scheme.Geophysical Research Letters, Vol. 17, No. 2.
Mufti, I. (1990). Large-scale three dimensional seismic models and their interpretative significance,Geophysics, Vol. 55, No. 9.
Piault, E., Duclos, P., and Sei, A. (1991). Gigaflops performance for 2D/3D heterogeneous acoustic modeling on a massively parallel computer.Proc. of the First Int'l. Conf. on the Math. and Numerical Aspects of Wave Propagation Phenomena, SIAM.
Reshef, M., Kosloff, D., Edwards, C., and Hsiung, C. (1988). Three dimensional acoustic modeling by Fourier method,Geophysics, Vol. 53, No. 9.
Sei, A. (1991). Etude de schemas numeriques pour des modèles de propagation d'ondes en milieu heterogene. Ph.D. Thesis, Université Paris IX-Dauphine.
Sei, A. (1993a). Computational cost of finite-difference elastic waves modeling.Proc. of the 63rd SEG Annual Meeting, Washington D.C.
Sei, A. (1994). Error analysis of numerical schemes for the wave equation in heterogeneous media.Applied Numerical Mathematics, Vol. 15.
Sei, A., and Collino, F. (1991). Discretization of the wave equation in Heterogeneous Media.Proc. of the First Int'l. Conf. on the Mathematical and Numerical Aspects of Wave Propagation Phenomena, SIAM.
Shubin, G. R., and Bell, J. B. (1987). A Modified equation approach to constructing fourth order methods for acoustic wave propagation.SIAM Journal on Scientific and Statistical Computing, Vol. 8.
Trefethen, L. (1982). Group velocity in finite difference schemes.SIAM Reviews, Vol. 24.
Vichnevetsky, R., and Bowles, J. B. (1982). Fourier analysis of numerical approximations of hyperbolic equations.SIAM Studies in Applied Mathematics, SIAM.
Whitham, G. B. (1974).Linear and Nonlinear waves. John Wiley & Sons.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sei, A., Symes, W. Dispersion analysis of numerical wave propagation and its computational consequences. J Sci Comput 10, 1–27 (1995). https://doi.org/10.1007/BF02087959
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02087959