Absorbing boundary conditions for difference approximations to the multidimensional wave equation

Higdon, Robert L.

doi:10.1090/s0025-5718-1986-0856696-4

Cited by 139 publications

(210 citation statements)

References 19 publications

(57 reference statements)

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“…Two main kinds of solutions have been proposed for this purpose: absorbing boundary conditions (ABCs) (e.g. Clayton and Engquist, 1977;Reynolds, 1978;Liao et al, 1984;Higdon, 1986;Higdon, 1991) and absorbing boundary layers (e.g. Cerjan et al, 1985;Kosloff and Kosloff, 1986;Compani-Tabrizi, 1986, Sochacki et al, 1987Bérenger, 1994;Komatitsch and Martin, 2007;Sen, 2010, 2012).…”

Section: Introductionmentioning

confidence: 99%

“…Collino (1993) devised a new scheme for high-order ABCs by using special auxiliary variables. This scheme is based on a reformulation of the sequence of ABC that was proposed by Higdon (1986). In contrast to the original formulation of the Higdon conditions, this scheme does not involve any high derivatives beyond the second order by introducing special auxiliary variables.…”

Section: Introductionmentioning

confidence: 99%

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Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Gao

Song

Zhang

et al. 2017

Exploration Geophysics

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Abstract. Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Gao

Song

Zhang

et al. 2017

Exploration Geophysics

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“…In the three-dimensional case, Higdon [25,26] showed it possible to allow the wave under an angle of incidence α with the outflow boundary. In the higher-order conditions even more angles can be chosen: α p , p = 1, .…”

Section: Three Dimensionsmentioning

confidence: 99%

Free-Surface Flow Simulations for Moored and Floating Offshore Platforms

Veldman

Luppes²,

Plas

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. During the development of the ComFLOW simulation method many challenges have to be tackled concerning the flow modelling and the numerical solution algorithm. Examples hereof are wave propagation, absorbing boundary conditions, fluid-solid body interaction, turbulence modeling and numerical efficiency. Some of these challenges will be discussed in the paper, in particular the design of absorbing boundary conditions and the numerical coupling for fluid-solid body interaction. As a demonstration of the progress made, a number of simulation results for engineering applications from the offshore industry will be presented: a wave-making oscillating buoy, a free-fall life boat dropping into wavy water, and wave impact against a semi-submersible offshore platform. For those applications, MARIN has carried out several validation experiments. 7515Arthur E.P. Veldman, et al.

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“…When fC j g 4 j¼1 are given by (40), what we need to prove is q 1 + q 2 = À(q 1 + q 2 )q 1 q 2 + 16q 1 q 2 . Following (29) and (30), we can write:…”

Section: Discrete Eigenvalue Problemmentioning

confidence: 99%

“…They can be said to occupy an intermediate position between the fully nonlocal ABCs and the local Sommerfeld conditions of type (11). In this category, one should first mention higher order local conditions by Bayliss and Turkel [25][26][27], as well as those by Higdon [28][29][30][31][32][33]. The first-order Sommerfeld conditions (11) can then be interpreted as the first member of the Bayliss-Turkel hierarchical sequence of boundary conditions.…”

Section: Sommerfeld Boundary Conditionsmentioning

confidence: 99%

Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions

Fibich

Tsynkov

2005

Journal of Computational Physics

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In [J. Comput. Phys. 171 (2001) 632-677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632-677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics.In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance, as well as in the range of physical phenomena that it is capable of simulating.

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Absorbing boundary conditions for difference approximations to the multidimensional wave equation

Cited by 139 publications

References 19 publications

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Free-Surface Flow Simulations for Moored and Floating Offshore Platforms

Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions

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