A fourth-order optimal finite difference scheme for the Helmholtz equation with PML

Dastour, Hatef; Liao, Wenyuan

doi:10.1016/j.camwa.2019.05.004

Cited by 21 publications

(22 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later, we shall present approximation of fourth order in accuracy. A straightforward computation with (6)- (17), the new difference scheme, is further given by…”

Section: A Fourth-order 21-point Finite Difference Schemementioning

confidence: 99%

“…Tables 3 and 4 give the numerical convergence and error of the new difference scheme for Example 2 and Example 3, respectively. Figures 8(a In studies [15,17], two noncompact 25-point difference schemes were proposed for the Helmholtz equation with the PML (perfectly matched layer) boundary condition. Both the two schemes are constructed based on the technique of derivative-weighting, and the former is of second order in accuracy while the latter is of fourth order.…”

Section: Examplementioning

confidence: 99%

“…In order to suppress the "pollution effect" and reduce the numerical dispersion, many numerical methods have been proposed in the past decades (cf. [1,2,[15][16][17][18][19]).…”

Section: Introductionmentioning

confidence: 99%

“…To reduce the numerical error, higher-order finite difference schemes (cf. [8,11,12,17,20]) were also commonly used. e higher-order schemes can be grouped into two types; one is compact and other one is noncompact.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

View full text Add to dashboard Cite

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.

show abstract

“…Later, we shall present approximation of fourth order in accuracy. A straightforward computation with (6)- (17), the new difference scheme, is further given by…”

Section: A Fourth-order 21-point Finite Difference Schemementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

“…In order to suppress the "pollution effect" and reduce the numerical dispersion, many numerical methods have been proposed in the past decades (cf. [1,2,[15][16][17][18][19]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

View full text Add to dashboard Cite

show abstract

“…For order 2 FDMs, [6,7] found that k 3 h 2 ≤ C (for some positive C independent of k, h) is required to obtain a reasonable solution. Meanwhile, for order 4 FDM, [9] found that k 5 h 4 ≤ C (for some positive C independent of k, h) is required to obtain a reasonable solution. From the previous discussion, it is clear that the grid size requirement for high order schemes is less stringent than low order ones.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Feng¹,

Han²,

Michelle³

2021

Preprint

View full text Add to dashboard Cite

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a sixth order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several stateof-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where kh is near 1 with k being the wavenumber and h the mesh size.

show abstract

A comparative study of several classes of meshfree methods for solving the Helmholtz equation

Liu,

et al. 2024

Engineering with Computers

View full text Add to dashboard Cite

A fourth-order optimal finite difference scheme for the Helmholtz equation with PML

Cited by 21 publications

References 25 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

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

A comparative study of several classes of meshfree methods for solving the Helmholtz equation

Contact Info

Product

Resources

About