A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions

1987 ◽  
Vol 13 (3) ◽  
pp. 221-234 ◽  
Author(s):  
Ronald F. Boisvert
Keyword(s):  
Helmholtz Equation ◽  
Fourth Order ◽  
Fourier Method ◽  
Three Dimensions

scholarly journals A comparison of finite-difference and fourier method calculations of synthetic seismograms

1989 ◽  
Vol 79 (4) ◽  
pp. 1210-1230
Author(s):  
C. R. Daudt ◽  
L. W. Braile ◽  
R. L. Nowack ◽  
C. S. Chiang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Group Velocity ◽  
Fourth Order ◽  
Difference Method ◽  
Fourier Method ◽  
Velocity Model ◽  
Second Order ◽  
Heterogeneous Model ◽  
Difference Calculation

Abstract The Fourier method, the second-order finite-difference method, and a fourth-order implicit finite-difference method have been tested using analytical phase and group velocity calculations, homogeneous velocity model calculations for disperson analysis, two-dimensional layered-interface calculations, comparisons with the Cagniard-de Hoop method, and calculations for a laterally heterogeneous model. Group velocity rather than phase velocity dispersion calculations are shown to be a more useful aid in predicting the frequency-dependent travel-time errors resulting from grid dispersion, and in establishing criteria for estimating equivalent accuracy between discrete grid methods. Comparison of the Fourier method with the Cagniard-de Hoop method showed that the Fourier method produced accurate seismic traces for a planar interface model even when a relatively coarse grid calculation was used. Computations using an IBM 3083 showed that Fourier method calculations using fourth-order time derivatives can be performed using as little as one-fourth the CPU time of an equivalent second-order finite-difference calculation. The Fourier method required a factor of 20 less computer storage than the equivalent second-order finite-difference calculation. The fourth-order finite-difference method required two-thirds the CPU time and a factor of 4 less computer storage than the second-order calculation. For comparison purposes, equivalent runs were determined by allowing a group velocity error tolerance of 2.5 per cent numerical dispersion for the maximum seismic frequency in each calculation. The Fourier method was also applied to a laterally heterogeneous model consisting of random velocity variations in the lower half-space. Seismograms for the random velocity model resulted in anticipated variations in amplitude with distance, particularly for refracted phases.

scholarly journals Reconstruction of the electric field of the Helmholtz equation in three dimensions

2017 ◽  
Vol 309 ◽  
pp. 56-78 ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Vo Anh Khoa ◽  
Mach Nguyet Minh ◽  
Thanh Tran
Keyword(s):  
Electric Field ◽  
Helmholtz Equation ◽  
Three Dimensions

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.

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