A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting

Cheng, Dapeng; Tan, Xu; Zeng, Taishan

doi:10.1016/j.camwa.2017.04.005

Cited by 18 publications

(12 citation statements)

References 26 publications

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“…Continue to expand (25) with i � ± 1, ± 2 at (m, n), and then substitute (25) and ( 26) into (7); we obtain…”

Section: Proposition 2 E 21-point Finite Difference Scheme (20) Is Of Fourth Order In Accuracymentioning

confidence: 99%

“…For solving the Helmholtz equation, we mainly have finite element methods (cf. [1][2][3][4][5]) and finite difference methods (see [6][7][8][9][10][11][12]). Due to the oscillatory characteristic of waves, all of the numerical methods are affected by the "pollution effect" (see [1,2]), which cannot be eliminated in two-dimensional (2D) and three-dimensional (3D) situations.…”

Section: Introductionmentioning

confidence: 99%

“…In [6], an optimal 9point difference scheme is developed, which is consistent with the Helmholtz equation with perfectly matched layer (PML). To improve the robustness, the authors of [7,16] established robust optimal difference schemes which works efficiently even if the step sizes along different directions are not equal. To reduce the numerical error, higher-order finite difference schemes (cf.…”

Section: Introductionmentioning

confidence: 99%

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An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

Cheng

Chen

Long

2021

Discrete Dynamics in Nature and Society

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In this paper, a new optimal fourth-order 21-point finite difference scheme is proposed to solve the 2D Helmholtz equation numerically, with the technique of matched interface boundary (MIB) utilized to treat boundary problems. For the approximation of Laplacian, two sets of fourth-order difference schemes are derived firstly based on the Taylor formula, with a total of 21 grid points involved. Then, a weighted combination of the two schemes is employed in order to reduce the numerical dispersion, and the weights are determined by minimizing the dispersion. Similarly, for the discretization of the zeroth-order derivative term, a weighted average of all the 21 points is implemented to obtain the fourth-order accuracy. The new scheme is noncompact; hence, it encounters great difficulties in dealing with the boundary conditions, which is crucial to the order of convergence. To tackle this issue, the matched interface boundary (MIB) method is employed and developed, which is originally used to accommodate free edges in the discrete singular convolution analysis. Convergence analysis and dispersion analysis are performed. Numerical examples are given for various boundary conditions, which show that new scheme delivers a fourth order of accuracy and is efficient in reducing the numerical dispersion as well.

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“…Continue to expand (25) with i � ± 1, ± 2 at (m, n), and then substitute (25) and ( 26) into (7); we obtain…”

Section: Proposition 2 E 21-point Finite Difference Scheme (20) Is Of Fourth Order In Accuracymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

Cheng

Chen

Long

2021

Discrete Dynamics in Nature and Society

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“…The Helmholtz equation finds a wide application in many fields of science, engineering, and industry. To solve the Helmholtz equation numerically, finite difference methods [1][2][3]) and finite element methods [4][5][6][7] are frequently employed. For its easy implementation and less computational complexity, the finite difference method is usually preferred, especially in the engineering field such as oil-gas exploration.…”

Section: Introductionmentioning

confidence: 99%

A Robust Optimal Finite Difference Scheme for the Three‐Dimensional Helmholtz Equation

Cheng

Chen

2019

Mathematical Problems in Engineering

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We propose a robust optimal 27-point finite difference scheme for the Helmholtz equation in three-dimensional domain. In each direction, a special central difference scheme with 27 grid points is developed to approximate the second derivative operator. The 27 grid points are divided into four groups, and each group is involved in the difference scheme by the manner of weighted combination. As for the approximation of the zeroth-order term, we use the weighted average of all the 27 points, which are also divided into four groups. Finally, we obtain the optimal weights by minimizing the numerical dispersion with the least-square method. In comparison with the rotated difference scheme based on a staggered-grid method, the new scheme is simpler, more practical, and much more robust. It works efficiently even if the step sizes along different directions are not equal. However, rotated scheme fails in this situation. We also present the convergence analysis and dispersion analysis. Numerical examples demonstrate the effectiveness of the proposed scheme.

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“…Moreover, a generalized optimal 9-point scheme for frequency-domain scalar wave equation was developed by Chen [13] which is an extension of the rotated 9-point scheme for the case that different spacial increments along x-axis and z-axis are used. Additionally, Cheng et al [18], in 2017, presented a new dispersion minimizing finite difference method in which the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy.…”

Section: Introductionmentioning

confidence: 99%

A generalized optimal fourth-order finite difference scheme for a 2D Helmholtz equation with the perfectly matched layer boundary condition

Dastour,

Liao

2019

Preprint

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A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference schemes, optimal point-weighting 25p and optimal pointweighting 17p. It is shown that the two schemes are consistent with the Helmholtz equation with PML. Moreover, an error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, we implement the refined choice strategy for selecting optimal parameters and present refined point-weighting 25p and refined pointweighting 17p finite difference schemes. Furthermore, three numerical examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion.

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A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting

Cited by 18 publications

References 26 publications

An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

A Robust Optimal Finite Difference Scheme for the Three‐Dimensional Helmholtz Equation

A generalized optimal fourth-order finite difference scheme for a 2D Helmholtz equation with the perfectly matched layer boundary condition

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