An efficient and high accuracy finite-difference scheme for the acoustic wave equation in 3D heterogeneous media

Li, Keran; Liao, Wenyuan

doi:10.1016/j.jocs.2019.101063

Cited by 15 publications

(13 citation statements)

References 25 publications

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“…ρ ∆u, which will be simpler to implement, where ∇u is approximated by the 4th-order compact finite difference scheme as above, and ∆u can be approximated by the 4th-order compact finite difference in [18]. Note that the approximation of the second spatial derivatives of u on the boundary requires a one-sided finite difference approximation similar to (2.13) and (2.14) with different coefficients.…”

Section: The New Compact Schemementioning

confidence: 99%

“…This approach, although being very efficient and compact, needs boundary conditions for u x and u xx , which are usually not available in the original problem. Under the smoothness assumption, a wave-equation based approach was proposed [18] to approximate the boundary conditions of u xx with arbitrary high order accuracy. It is worth to mention that many high-order compact finite difference methods are only available for constant coefficients cases and fail in variable coefficients cases.…”

Section: Introductionmentioning

confidence: 99%

“…To address this issue, new schemes have been proposed. Recently, two 4th-order compact finite difference schemes for acoustic wave equations with variable sound speed was introduced in [18,19]. In [4], an energy norm based method was applied to analyze the stability of the finite difference methods for acoustic wave equation with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…The methods proposed in [18,19] work very effectively for variable acoustic velocity with constant media density cases, however, many real-world applications deal with the variable media density case, for which the Laplacian operator in spatial dimension is replaced by the divergence form given by ∇ • 1 ρ ∇u . It has also been shown [7] that the fourth-order accuracy can be obtained only if the density is smooth.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient and Stable Finite Difference Modelling of Acoustic Wave Propagation in Variable-density Media

Li¹,

Li²,

Liao³

2020

Preprint

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In this paper, we consider the development and analysis of a new explicit compact high-order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth-order accuracy in space and second-order accuracy in time. The compactness of the scheme is obtained by the so-called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one-sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Currant-Friedrichs-Lewy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with Perfect Matched Layer boundary condition is also validated in this paper as well.

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Section: The New Compact Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient and Stable Finite Difference Modelling of Acoustic Wave Propagation in Variable-density Media

Li¹,

Li²,

Liao³

2020

Preprint

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“…More recent developments on explicit two‐step finite difference schemes, particularly on high‐order schemes, are reviewed in References 8-11. Some related issues of computational complexity of explicit finite difference schemes are discussed, for example, in References 12,13.…”

Section: Introductionmentioning

confidence: 99%

Explicit stencil computation schemes generated by Poisson's formula for the 2D wave equation

Khutoryansky

2020

Engineering Reports

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A new approach to building explicit time-marching stencil computation schemes for the transient two-dimensional acoustic wave equation is implemented. It is based on using Poisson's formula and its three time level modification combined with polynomial stencil interpolation of the solution at each time-step and exact integration. The time-stepping algorithm consists of two explicit stencil computation procedures: a first time-step procedure incorporating the initial conditions and a two-step scheme for the second and next time-steps. Three particular explicit stencil schemes (with 5, 9, and 13 space points) are constructed using this approach. Their stability regions are presented. All of the obtained first time-step computation expressions are different from those used in conventional finite-difference methods. Accuracy advantages of the new schemes in comparison with conventional finite-difference schemes are demonstrated by simulation using an exact benchmark solution. K E Y W O R D S 2D wave equation, first time-step, stencil computation, two-step scheme INTRODUCTIONStencil computations are widely implemented in many numerical algorithms that involve structured grids. In acoustic field simulation based on two-(2D) or three-dimensional transient wave equations, the finite difference time domain method is a standard approach leading to stencil operations. 1 Finite difference schemes for the 2D wave equation have been extensively employed in seismic simulations and linear modeling of vibrations of membranes. In an explicit time-stepping finite difference scheme, a solution value at each point in a time-space grid is calculated using a linear combination of values at its spatial neighbors from previous time steps. Among such schemes the main attention in the literature has been given to two-step schemes (which operate over three time levels t k+1 = (k + 1) , t k = k and t k−1 = (k − 1) where is a fixed time increment). They have been intensively studied and described in many articles and books (eg, References 2-7). More recent developments on explicit two-step finite difference schemes, particularly on high-order schemes, are reviewed in References 8-11. Some related issues of computational complexity of explicit finite difference schemes are discussed, for example, in References 12,13. Explicit two-step numerical schemes for the scalar wave equation can also be devised based on spherical means representations as it is done in Reference 14 where an integral evolution formula with three time levels is derived. For the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media

Liao

2023

Numerical Methods Partial

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In this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.

show abstract

An efficient and high accuracy finite-difference scheme for the acoustic wave equation in 3D heterogeneous media

Cited by 15 publications

References 25 publications

Efficient and Stable Finite Difference Modelling of Acoustic Wave Propagation in Variable-density Media

Efficient and Stable Finite Difference Modelling of Acoustic Wave Propagation in Variable-density Media

Explicit stencil computation schemes generated by Poisson's formula for the 2D wave equation

A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media

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