scholarly journals Optimal 25-Point Finite-Difference Subgridding Techniques for the 2D Helmholtz Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Tingting Wu ◽  
Zhongying Chen ◽  
Jian Chen
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
Helmholtz Equation ◽  
Numerical Experiments ◽  
Numerical Dispersion ◽  
Computational Domain ◽  
Seismic Modeling ◽  
Finite Difference Equation ◽  
Choice Strategy ◽  
Interior Domain

We present an optimal 25-point finite-difference subgridding scheme for solving the 2D Helmholtz equation with perfectly matched layer (PML). This scheme is second order in accuracy and pointwise consistent with the equation. Subgrids are used to discretize the computational domain, including the interior domain and the PML. For the transitional node in the interior domain, the finite difference equation is formulated with ghost nodes, and its weight parameters are chosen by a refined choice strategy based on minimizing the numerical dispersion. Numerical experiments are given to illustrate that the newly proposed schemes can produce highly accurate seismic modeling results with enhanced efficiency.

Sur l'existence d'une longueur élémentaire et d'un intervalle de temps élémentaire

1978 ◽  
Vol 56 (8) ◽  
pp. 1109-1115 ◽  
Author(s):  
Robert Lacroix
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
One Dimension ◽  
Time Interval ◽  
Finite Difference Equation ◽  
Elementary Length

We have briefly examined several studies which have been made concerning the introduction of an elementary length l0 and an elementary time interval t0 into physical theories. We have discussed the arguments which we have found, arguments formulated by other authors, and which support the hypotheses concerning the existence of l0 and of t0. A finite difference equation is proposed and the solutions of some problems of movement in one dimension are given.

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.

ON SOLVING NONLINEAR FUNCTIONAL, FINITE DIFFERENCE, COMPOSITION, AND ITERATED EQUATIONS

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 277-282 ◽  
Author(s):  
RAY BROWN
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
Difference Equations ◽  
Finite Difference Equation ◽  
Euler Approximation ◽  
Nonlinear Functional ◽  
Finite Difference Equations ◽  
Wide Range ◽  
Iterated Equations ◽  
General Method

In this letter, we present a general method for solving a wide range of nonlinear functional and finite difference equations, as well as iterated equations such as the Hénon and Mandelbrot equations. The method extends to differential equations using an Euler approximation to obtain a finite difference equation.

Accurate and efficient seismic modeling in random media

Geophysics ◽  
1993 ◽  
Vol 58 (4) ◽  
pp. 576-588 ◽  
Author(s):  
Guido Kneib ◽  
Claudia Kerner
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Random Media ◽  
Polynomial Interpolation ◽  
Random Medium ◽  
High Order ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Seismic Modeling

The optimum method for seismic modeling in random media must (1) be highly accurate to be sensitive to subtle effects of wave propagation, (2) allow coarse sampling to model media that are large compared to the scale lengths and wave propagation distances which are long compared to the wavelengths. This is necessary to obtain statistically meaningful overall attributes of wavefields. High order staggered grid finite‐difference algorithms and the pseudospectral method combine high accuracy in time and space with coarse sampling. Investigations for random media reveal that both methods lead to nearly identical wavefields. The small differences can be attributed mainly to differences in the numerical dispersion. This result is important because it shows that errors of the numerical differentiation which are caused by poor polynomial interpolation near discontinuities do not accumulate but cancel in a random medium where discontinuities are numerous. The differentiator can be longer than the medium scale length. High order staggered grid finite‐difference schemes are more efficient than pseudospectral methods in two‐dimensional (2-D) elastic random media.

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