On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids

Gao, Longfei; Ketcheson, David I.; Keyes, David E.

doi:10.1093/gji/ggx470

Cited by 4 publications

(10 citation statements)

References 22 publications

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“…such that the wave can pass through the interface smoothly without interference. In this section, we explain how to impose the interface conditions in (18) with the SATs. In order to focus on the interface treatment, we assume that both boundaries (x L and x R ) have been satisfactorily dealt with by some SBP discretization operators and SATs in an energy-conserving manner so that in the upcoming energy analysis, no term pops out in Q due to the boundaries.…”

Section: Imposing Interface Conditions For the 1d Model Problemmentioning

confidence: 99%

SBP–SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids

Gao

Fernández

Carpenter

et al. 2019

Journal of Computational and Applied Mathematics

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We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming interfaces. In doing so, variations in the wave speeds of the subterranean media can be accounted for more efficiently. Staggered grid finite difference operators satisfying the summation-by-parts (SBP) property are devised to approximate the spatial derivatives appearing in the acoustic wave equation. These operators are applied within each block independently. The coupling between blocks is achieved through simultaneous approximation terms (SATs), which impose the interface condition weakly, i.e., by penalty. Ratio of the grid spacing of neighboring blocks is allowed to be rational number, for which specially designed interpolation formulas are presented. These interpolation formulas constitute key pieces of the simultaneous approximation terms. The overall discretization is shown to be energy-conserving and examined on test cases of both theoretical and practical interests, delivering accurate and stable simulation results.

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Section: Imposing Interface Conditions For the 1d Model Problemmentioning

confidence: 99%

SBP–SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids

Gao

Fernández

Carpenter

et al. 2019

Journal of Computational and Applied Mathematics

Self Cite

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“…Left multiplying (42) with ḃT and recalling the definitions of the discrete energies in (20), we have the following result:…”

Section: Interface Treatmentmentioning

confidence: 99%

“…regarding the discrete energy associated with (42) in the FEM region. When it comes to implementation, the line integral ∂I in ( 43) is usually replaced by quadrature.…”

Section: Interface Treatmentmentioning

confidence: 99%

“…In previous subsections, we have described in detail the spatial discretizations in the FEM region and the FDM region, as well as the interface treatment, resulting in the semi-discretized systems (42) and (49). In the following, we explain how these semi-discretized systems can be numerically integrated in time.…”

Section: Full Discretizationmentioning

confidence: 99%

“…Updating procedures for semi-discretized systems in the FDM region and the FEM region, i.e., (49) and(42), respectively.…”

mentioning

confidence: 99%

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Combining finite element and finite difference methods for isotropic elastic wave simulations in an energy-conserving manner

Gao

Keyes

2019

Journal of Computational Physics

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We consider numerical simulation of the isotropic elastic wave equations arising from seismic applications with non-trivial land topography. The more flexible finite element method is applied to the shallow region of the simulation domain to account for the topography, and combined with the more efficient finite difference method that is applied to the deep region of the simulation domain. We demonstrate that these two discretization methods, albeit starting from different formulations of the elastic wave equation, can be joined together smoothly via weakly imposed interface conditions. Discrete energy analysis is employed to derive the proper interface treatment, leading to an overall discretization that is energy-conserving. Numerical examples are presented to demonstrate the efficacy of the proposed interface treatment.

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Direct implementation of discontinuous-grid finite-difference method using multiple point sources method and dynamic wavefield injection

Miao

Zhang

2023

Geophysical Journal International

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Summary The classical finite-difference (FD) method stimulates wave propagation in uniform grids. In models with considerable velocity variations, the computational efficiency is compromised by oversampling in time and space. Although adopting a discontinuous grid in different wave-speed regions can improve computational efficiency, such a technique is typically hindered by low accuracy in transition zones (i.e. in the vicinity of the discontinuous-grid interface) and is typically unstable in the long term. We propose a direct implementation of the discontinuous-grid FD method by performing two simulations simultaneously: one on a coarse grid and the other on a fine grid. The proposed method applies a dynamic injection strategy to manage wavefield communication between coarse and fine submodels. Compared with previous discontinuous-grid FD methods, where the number of layers required for wavefield communication is one-half that the order of the FD scheme, the proposed method only requires one single layer, which significantly reduces the communication overhead and suppresses wavefield errors. Numerical experiments show that the results yielded by our method are consistent with the reference solutions yielded by the uniform-grid FD method via a fine grid. Furthermore, our method does not encounter numerical instability in long-term simulations. Therefore, the proposed discontinuous-grid FD method can accelerate numerical simulations while retaining stable numerical accuracy, even for long-term simulations.

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On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids

Cited by 4 publications

References 22 publications

SBP–SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids

SBP–SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids

Combining finite element and finite difference methods for isotropic elastic wave simulations in an energy-conserving manner

Direct implementation of discontinuous-grid finite-difference method using multiple point sources method and dynamic wavefield injection

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