scholarly journals The Best Finite-Difference Scheme for the Helmholtz Equation

2012 ◽  
Vol 02 (03) ◽  
pp. 207-212 ◽  
Author(s):  
T. Zhanlav ◽  
V. Ulziibayar
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Helmholtz Equation ◽  
Finite Difference Scheme

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

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Dongsheng Cheng ◽  
Jianjun Chen ◽  
Guangqing Long
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Helmholtz Equation ◽  
Fourth Order ◽  
Finite Difference Scheme ◽  
Weighted Average ◽  
Numerical Dispersion ◽  
Taylor Formula ◽  
Interface Boundary

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.

scholarly journals A Robust Optimal Finite Difference Scheme for the Three-Dimensional Helmholtz Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Dongsheng Cheng ◽  
Baowen Chen ◽  
Xiangling Chen
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Helmholtz Equation ◽  
Finite Difference Scheme ◽  
Three Dimensional ◽  
Weighted Average ◽  
Least Square ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Grid Points

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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