Well-posedness of the diffusive-viscous wave equation arising in geophysics

Han, Weimin; Gao, Jinghuai; Zhang, Yijie; Xu, Wenhao

doi:10.1016/j.jmaa.2020.123914

Cited by 14 publications

(6 citation statements)

References 6 publications

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“…The existence and uniqueness of the solution of the DVWE in a bounded domain with mixed boundary conditions are given in [11]. However, there is no result regarding the well-posedness of the diffusive-viscous wave equation in the unbounded domain.…”

Section: Preliminarymentioning

confidence: 99%

“…Then the estimate (12) holds by integrating the above equation from 0 to t and using the estimate (11).…”

Section: Existence and Uniqueness Of The Weak Solutionmentioning

confidence: 99%

“…We are now able to show the existence and uniqueness of the continuous problem (8). We use more or less the standard compactness arguments similar as that used in [11] based on the properties of the Hermite spectral Galerkin approximation.…”

Section: Existence and Uniqueness Of The Weak Solutionmentioning

confidence: 99%

“…Comparing the above equation with ( 23), we have ∂ t u(0) = w 0 . The estimate (15) follows by letting N → ∞ in (11).…”

Section: Existence and Uniqueness Of The Weak Solutionmentioning

confidence: 99%

“…However, most existing works only focus on the numerical schemes and the corresponding results of the stability and error estimates, there is few theoretical work concerning the existence and uniqueness of the solution for DVWEs. Most recently, we noticed an important work provided by Han et al in [11], in which the well-posedness and stability of DVWEs were established in bounded domains, this provided a theoretical foundation to develop numerical methods. However, no regularity result is discussed in the work.…”

Section: Introductionmentioning

confidence: 98%

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Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysics

Liu¹,

Mao²

2022

Preprint

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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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Section: Preliminarymentioning

confidence: 99%

“…Then the estimate (12) holds by integrating the above equation from 0 to t and using the estimate (11).…”

Section: Existence and Uniqueness Of The Weak Solutionmentioning

confidence: 99%