High-order finite difference methods for the Helmholtz equation

Singer, Ivan; Turkel, Eli

doi:10.1016/s0045-7825(98)00023-1

Cited by 161 publications

(150 citation statements)

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“…The discretization that is particularly of interest in this work is the O(h 4 ) accurate discretization based on the Padé approximation. It is called the HO discretization in [29], with stencil,…”

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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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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“…Solving these equations we find that the values of c + and c − are the same as in the formula for u F −pml (11), and…”

Section: The Modified Equationmentioning

confidence: 84%

A perfectly matched layer for the Helmholtz equation in a semi-infinite strip

Singer

Turkel

2004

Journal of Computational Physics

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The Perfectly Matched Layer (PML) has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains. We develop a discretization to solve the Helmholtz equation in an infinite two dimensional strip. We solve the interior equation using high-order finite differences schemes. The combined Helmholtz-PML problem is then analyzed for the parameters that give the best performance. We show that the use of local high-order methods in the physical domain coupled with a specific second order approximation in the PML yields global high-order accuracy in the physical domain. We discuss the impact of the parameters on the effectiveness of the PML. Numerical results are presented to support the analysis.keywords.

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“…In this paper we only consider a zero right hand side. In [13] the formulae for including the right hand is given. Let φ i,j be a numerical approximation to v (x i , y j ).…”

Section: Discretizationmentioning

confidence: 99%

“…This requirement is, however, less demanding if a higher order discretization is used. For finite differences, examples of high order accurate finite-difference discretizations on uniform meshes are introduced and analyzed by Harari and Turkel [12], and later by Singer and Turkel [13,14].…”

Section: Introductionmentioning

confidence: 99%

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation

Erlangga

Turkel

2012

ESAIM: M2AN

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Abstract. We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges slowly, a preconditioner is introduced, which is a Helmholtz equation but with a modified complex wavenumber. This is discretized by a second or fourth order compact scheme. The system is solved by BICGSTAB with multigrid used for the preconditioner. We study, both by Fourier analysis and computations this preconditioned system especially for the effects of high order discretizations.Mathematics Subject Classification. 35J47, 65M06.

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High-order finite difference methods for the Helmholtz equation

Cited by 161 publications

References 5 publications

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

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

A perfectly matched layer for the Helmholtz equation in a semi-infinite strip

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation

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