Abstract
The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.
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Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
D. Ling would like to acknowledge support by National Natural Science Foundation of China grant 12101486, China Postdoctoral Science Foundation grant 2020M683446 and the High-performance Computing Platform at Xi’an Jiaotong University. Z. Mao would like to acknowledge support by the Fundamental Research Funds for the Central Universities (20720210037).
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Ling, D., Mao, Z. Analysis and Hermite Spectral Approximation of Diffusive-Viscous Wave Equations in Unbounded Domains Arising in Geophysics. J Sci Comput 95, 51 (2023). https://doi.org/10.1007/s10915-023-02175-9
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DOI: https://doi.org/10.1007/s10915-023-02175-9
Keywords
- Diffusive-viscous wave equations
- Well-posedness
- Regularity
- Unbounded domain
- Artificial reflection
- Error estimates