A modified wave equation with diffusion effects and its application as a smoothing scheme for seismic wave propagation simulations. (English) Zbl 1499.65394
Summary: We present a new modified wave equation and apply it to develop a smoothing scheme for seismic wave propagation simulations. With mathematical rigour we show that the solution of the new equation, which is derived as an analog of the advection-diffusion equation, can be obtained by the spatial convolution between a solution of the wave equation and the heat kernel and has a finite propagation speed and a diffusion effect. Using numerical experiments we show that the smoothing scheme based on the modified wave equation has the following advantages. Firstly, it preserves the characteristics of the wave equation such as wave propagation speed. Secondly, it selectively removes the short-wavelength components of the solution. Lastly, the energy decreases slowly after the short-wavelength components have been removed. Since our smoothing scheme can be implemented by adding simple correction terms to usual schemes, it can easily be applied to the seismic wave equation.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35Q86 | PDEs in connection with geophysics |
| 86A15 | Seismology (including tsunami modeling), earthquakes |
Keywords:
wave equation; diffusion effects; smoothing scheme; seismic wave equation; long-period ground motion evaluationReferences:
| [1] | Aoi, S.; Fujiwara, H., 3-D finite difference method using discontinuous grids, Bull. Seismol. Soc. Amer., 89, 918-930 (1999) |
| [2] | Brezis, H., Functional Analysis, Sobolev Spacecs and Partial Differential Equations (2010), Springer: Springer, New York · Zbl 1218.46002 |
| [3] | Cerjan, C.; Kosloff, D.; Kosloff, R.; Reshef, M., A nonreflecting boundary condition for discrete acoustic and elastic wave equations, Geophysics, 50, 705-708 (1985) · doi:10.1190/1.1441945 |
| [4] | Daftardar-Gejji, V.; Bhalekar, S., Solving fractional diffusion-wave equations using a new iterative method, Fract. Calc. Appl. Anal., 11, 2, 193-202 (2008) · Zbl 1210.26009 |
| [5] | Delic, A., Fractional in time diffusion-wave equation and its numerical approximation, Filomat, 30, 5, 1375-1385 (2016) · Zbl 1474.35642 · doi:10.2298/FIL1605375D |
| [6] | Folland, G. B., Introduction to Partial Differential Equations (1995), Princeton University Press: Princeton University Press, Princeton, NJ · Zbl 0841.35001 |
| [7] | Freziger, J. H., Computational Methods for Fluid Dynamics (2001), Springer: Springer, Berlin |
| [8] | Gao, L.; Brossier, R.; Virieux, J., Using time filtering to control the long-time instability in seismic wave simulation, Geophys. J. Int., 204, 1443-1461 (2016) · doi:10.1093/gji/ggv534 |
| [9] | Garg, M.; Manohar, P., Numerical solution of fractional diffusion-wave equation with two space variables by matrix method, Frac. Calc. Appl. Anal., 13, 2, 192-207 (2010) · Zbl 1211.26004 |
| [10] | John, F., Partial Differential Equations (1982), Springer: Springer, New York · Zbl 0472.35001 |
| [11] | Maeda, T.; Morikawa, N.; Iwaki, A.; Aoi, S.; Fujiwara, H., Finite-difference simulation of long-period ground motion for the Nankai Trough megathrust earthquakes, J. Disaster Res., 8, 5, 912-925 (2013) · doi:10.20965/jdr.2013.p0912 |
| [12] | Mitkowski, W., Approximation of fractional diffusion-wave equation, Acta Mech. Automat., 5, 2, 65-68 (2011) |
| [13] | Patankar, S. V., Numerical Heat Transfer and Fluid Flow (1980), Hemisphere Publishing Corporation: Hemisphere Publishing Corporation, Washington, DC · Zbl 0521.76003 |
| [14] | Preston, L.A., Aldridge, D.F., and Symons, N.P., Finite-difference modeling of 3D seismic wave propagation in high-contrast media, SEG Technical Programs Expanded Abstracts, 2008, pp. 2142-2146. |
| [15] | Schwartz, L., Mathematics for The Physical Sciences (1966), Hermann: Hermann, Paris · Zbl 0151.34001 |
| [16] | Takewaki, H.; Yabe, T., The cubic-interpolated pseudo particle (CIP) method: Application to nonlinear and multi-dimensional hyperbolic equations, J. Comput. Phys., 70, 355-372 (1987) · Zbl 0624.65079 · doi:10.1016/0021-9991(87)90187-2 |
| [17] | Yabe, T.; Aoki, T., A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver, Comput. Phys. Commun., 66, 219-232 (1991) · Zbl 0991.65521 · doi:10.1016/0010-4655(91)90071-R |
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