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Absorbing boundary conditions for wave propagation in viscoelastic media. (English) Zbl 0864.73020

Summary: We derive absorbing boundary conditions for viscoelastic waves. Error estimates are given for such absorbing boundary conditions. We treat the problem in the space-frequency domain formulation after taking the Fourier transformation in time. Well-posedness is shown for the elliptic problems thus created, and Fourier invertibility is also proved. We also present some results of numerical simulations.

MSC:

74J10 Bulk waves in solid mechanics
74Hxx Dynamical problems in solid mechanics
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[1] Adams, R. A., Sobolev Spaces (1975), Academic press: Academic press London · Zbl 0186.19101
[2] Brennan, B. J.; Smylie, D. E., Linear viscoelasticity and dispersion in seismic wave propagation, Rev. Geophys. Space Phys., 19, 233-246 (1981)
[3] Carcione, J. M.; Kosloff, D.; Kosloff, R., Viscoelastic wave propagation simulation in the earth, Geophysics, 53, 769-777 (1988) · Zbl 0656.73020
[4] Christensen, R. M., Theory of Viscoelasticity: An Introduction (1971), Academic press: Academic press New York
[5] Clayton, R.; Engquist, B., Absorbing boundary conditions for acoustic and elastic wave equations, Bull. Seismol. Soc. Amer., 67, 1529-1540 (1977)
[6] Douglas, J.; Santos, J. E.; Sheen, D., Approximation of scalar waves in the space-frequency domain, Math. Models Methods Appl. Sci., 4, 509-531 (1994) · Zbl 0812.35173
[7] Douglas, J.; Santos, J. E.; Sheen, D.; Bennethum, L. S., Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci., 3, 171-194 (1993) · Zbl 0783.65070
[8] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31, 629-651 (1977) · Zbl 0367.65051
[9] Engquist, B.; Majda, A., Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math., 32, 313-357 (1979) · Zbl 0387.76070
[10] Flügge, W., Viscoelasticity (1975), Springer: Springer New York · Zbl 0352.73033
[11] Golden, J. M.; Graham, G. A.C., Boundary Value Problems in Linear Viscoelasticity (1988), Springer: Springer New York · Zbl 0651.73011
[12] Higdon, R. L., Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation, Math. Comput., 47, 437-459 (1986) · Zbl 0609.35052
[13] Higdon, R. L., Radiation boundary conditions for elastic wave propagation, SIAM J. Numer. Anal., 27, 831-870 (1990) · Zbl 0718.35058
[14] Liu, H. P.; Anderson, D. L.; Kanamori, H., Velocity dispersion due to anelasticity; implications for seismology and mental composition, Geophys. J. Roy. Astron. Soc., 47, 41-58 (1976)
[15] MacCamy, R. C., Absorbing boundaries for viscoelasticity, (Viscoelasticity and Rheology (1986), Academic Press: Academic Press New York), 323-344 · Zbl 0587.73049
[16] Tal-Ezer, H.; Carcione, J. M.; Kosloff, D., An accurate and efficient scheme for wave propagation in linear viscoelastic media, Geophysics., 55, 1366-1379 (1990)
[17] Tscholegl, N. W., The Phenomenological Theory of Linear Viscoelastic Behavior (1989), Springer: Springer New York
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