Second-Order Absorbing Boundary Conditions for the Wave Equation: A Solution for the Corner Problem

Bamberger, A.; Joly, Patrick; Roberts, Jean E.

doi:10.1137/0727021

Cited by 83 publications

(92 citation statements)

References 11 publications

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“…Earlier work was done by Bamberger et al [28] and Collino [29]. These approaches formulate the corner absorbers by deriving corner compatibility conditions, which are necessary to ensure the regularity of the solution of the problem when the data are smooth.…”

Section: Extension To Corner Regionsmentioning

confidence: 99%

“…This truncation introduces error in the half-space stiffness, which can be measured with the help of the reflection coefficient (the ratio between the reflected and incident wave amplitudes at the computational boundary). For the sake of simplified presentation, the reflection coefficient is derived later in this section and is given by (28). Based on (28), note that perfect absorption is achieved when k x = 2i/L j .…”

Section: Continued Fraction Absorbing Boundary Conditionsmentioning

confidence: 99%

“…For the sake of simplified presentation, the reflection coefficient is derived later in this section and is given by (28). Based on (28), note that perfect absorption is achieved when k x = 2i/L j . This indicates that the mesh in Figure 3 is effective for evanescent waves, but completely reflects the propagating waves since |R| ≡ 1 for all real values of k x .…”

Section: Continued Fraction Absorbing Boundary Conditionsmentioning

confidence: 99%

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Continued fraction absorbing boundary conditions for convex polygonal domains

Guddati

Lim

2005

Numerical Meth Engineering

100

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SUMMARYContinued fraction absorbing boundary conditions (CFABCs) are highly effective boundary conditions for modelling wave absorption into unbounded domains. They are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries. In this paper, CFABCs are extended to the more general case of polygonal computational domains. The key to the current development is the surprising link found between the CFABCs and the complex co-ordinate stretching of perfectly matched layers (PMLs). This link facilitates the extension of CFABCs to oblique corners and, thus, to polygonal domains. It is shown that the proposed CFABCs are easy to implement, expected to perform better than PMLs, and are effective for general polygonal computational domains. In addition to the derivation of CFABCs, a novel explicit time-stepping scheme is developed for efficient numerical implementation. Numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain. Although this paper focuses on wave propagation, its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients.

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Section: Extension To Corner Regionsmentioning

confidence: 99%

Section: Continued Fraction Absorbing Boundary Conditionsmentioning

confidence: 99%

Section: Continued Fraction Absorbing Boundary Conditionsmentioning

confidence: 99%

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Continued fraction absorbing boundary conditions for convex polygonal domains

Guddati

Lim

2005

Numerical Meth Engineering

100

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“…At Γ e , this equation is supplemented by second-order absorbing boundary conditions, as described in [2,8]. 2.2.…”

Section: Mathematical Problem Definitionmentioning

confidence: 99%

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

Umetani

MacLachlan

Oosterlee

2009

Numerical Linear Algebra App

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Abstract. In this paper, an iterative solution method for a fourth-order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in [10], where multigrid was employed as a preconditioner for a Krylov subspace iterative method. This multigrid preconditioner is based on the solution of a second Helmholtz operator with a complexvalued shift. In particular, we compare preconditioners based on a point-wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in [37] with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi-CGSTAB with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous high-wavenumber problems.Key words. Helmholtz equation, non-constant high wavenumber, complex-valued multigrid preconditioner, algebraic multigrid, ILU smoother AMS subject classifications. 65N55, 65F10, 78A45, 35J051. Introduction. Many authors, e.g. [5,7,13,19], have contributed to the development of appropriate multigrid methods for the Helmholtz equation, but an efficient multigrid treatment of heterogeneous problems with high wavenumers arising in engineering settings has not yet been proposed in the literature. The multigrid method [4,14] is known to be a highly efficient iterative method, for example, for discrete Poisson-type equations, even with fourth-order accurate discretizations [6,33]. The Helmholtz equation, however, does not belong to the class of PDEs for which off-the-shelf multigrid methods perform efficiently. Convergence degradation and, consequently, loss of O(N ) complexity are caused by difficulties encountered in the smoothing and coarse-grid correction components; see [7,33] for a discussion.We present an efficient numerical solution technique for the heterogeneous highwavenumber Helmholtz equation, discretized by fourth-order finite differences. Recently,in [10], a robust preconditioned Bi-CGSTAB method has been proposed for solving these problems, in which the preconditioner is based on a second Helmholtz equation with an imaginary shift. This preconditioner is a member of the family of shifted Laplacian operators, introduced in [20], and its inverse can be efficiently approximated by means of a multigrid iteration. Two-dimensional results, representative for geophysical applications, generated by second-order finite differences, have been presented in [26] and 3D results in [27].In this paper, we generalize this solver and include, in particular, a fourth-order discretization of the Helmholtz operator in our discussion. The multigrid preconditioner is enhanced, in the sense that we replace the point-wise Jacobi smoother in the multigrid preconditioner by a variant of the incomplete lower-upper factorization smoother, ILU(0). Furthermore, we evaluate the performance of a prolongation

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“…If the solution of ( 1.1 ) is smooth enough, an application of equation ( (1.2.iv) utt + uty -\uxx = 0 onY\xJ, which is given by Engquist and Majda [5], and earlier proposed by Claerbout [2]. Indeed, Engquist and Majda [6] and Bamberger, Joly, and Roberts [1] suggested modified forms of (1.2) to take into account corner instability. However, in this paper we shall use the boundary condition (1.2) or (l.l.ii) for simplicity.…”

Section: Introductionmentioning

confidence: 99%

Second-order absorbing boundary conditions for the wave equation in a rectangular domain

Sheen¹

1993

Math. Comp.

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Abstract. We study finite element methods for the wave equation in a rectangular domain with a second-order absorbing boundary condition imposed on the boundary. For this problem there seems to be no known finite element method, although many finite difference methods have been proposed. A thirdorder energy, however, will be introduced which will be utilized to reduce our original second-order problem to a first-order symmetric dissipative hyperbolic system. Then, for this first-order system a weak formulation will be given and continuous-time and discrete-time Galerkin procedures will be investigated. Error estimates will also be given.

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Second-Order Absorbing Boundary Conditions for the Wave Equation: A Solution for the Corner Problem

Cited by 83 publications

References 11 publications

Continued fraction absorbing boundary conditions for convex polygonal domains

Continued fraction absorbing boundary conditions for convex polygonal domains

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

Second-order absorbing boundary conditions for the wave equation in a rectangular domain

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