Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number

Turkel, Eli; Gordon, Dan; Gordon, Rachel; Tsynkov, Semyon

doi:10.1016/j.jcp.2012.08.016

Cited by 132 publications

(103 citation statements)

References 21 publications

(43 reference statements)

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“…2 Adaptive methods, on the other hand, aim to leverage à priori knowledge of the solution of the Helmholtz equation, such as its known oscillatory behavior. In practice, adaptive methods have mostly focused on adaptivity to the medium, such as polynomial Galerkin methods with hp refinement [3,70,73,96,107,111], specially optimized finite differences [23,45,92,93,102] and finite elements [4,99], enriched finite elements [30][31][32][33], plane wave methods [5,21,42,43,46,69,74], generalized plane wave methods [54,55], locally corrected finite elements [17,38,82], and discretizations with specially chosen basis functions [7,8,76], among many others. They have been especially successful on reducing the pollution effect by accurately capturing the dispersion relation.…”

Section: Introductionmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

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Section: Introductionmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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“…The Helmholtz equation has important applications in acoustic and electromagnetic waves. Obtaining an efficient and more accurate numerical solution for the Helmholtz equation has always been a hot topic in wave computation (see [1][2][3][4][5][6] and the reference therein).…”

Section: Introductionmentioning

confidence: 99%

A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equation

2017

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper, we present a dispersion minimizing compact finite difference scheme for solving the 2D Helmholtz equation, which is a fourth-order scheme. The error between the numerical wavenumber and the exact wavenumber is analyzed, and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters. Numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters.

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“…In an effort to develop high-order methods, compact finite difference schemes are proposed for various problems [9,19,23,24]. FD methods have low geometric flexibility, in particular, for high-order schemes.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Spectral Difference Methods for an Elliptic Equation

Jeon

Park

Shin

2017

Computational Methods in Applied Mathematics

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A locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, resulting in the system of equations in the cell interface unknowns only. A complete ellipticity analysis is provided. The optimal order of convergence in the semi-discrete energy norms is proved. Several numerical results are given to show the performance of the method, which confirm our theoretical findings.

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Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number

Cited by 132 publications

References 21 publications

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equation

Hybrid Spectral Difference Methods for an Elliptic Equation

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