Advances in Radio Science Optimally Accurate Second-Order Time-Domain Finite-Difference Scheme for Acoustic, Electromagnetic, and Elastic Wave Modeling (original) (raw)
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The Finite-Difference Time-Domain Method for Modeling of Seismic Wave Propagation
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We present a review of the recent development in finite-difference time-domain modeling of seismic wave propagation and earthquake motion. The finite-difference method is a robust numerical method applicable to structurally complex media. Due to its relative accuracy and computational efficiency it is the dominant method in modeling earthquake motion and it also is becoming increasingly more important in the seismic industry and for structural modeling. We first introduce basic formulations and properties of the finite-difference schemes including promising recent advances. Then we address important topics as material discontinuities, realistic attenuation, anisotropy, the planar free surface boundary condition, free-surface topography, wavefield excitation (including earthquake source dynamics), nonreflecting boundaries, and memory optimization and parallelization. 0 J y y J y Δ = + , 0 K z z K z Δ = + , , , , u I J K m or , , m I J K u , is approximated by a grid function , , ( , , , ) I J K I J K I J K I J K I I-1/2 J +3/2 J +1 J J J -1/2 h/2 I+1 I J +3/2 J +1 J J J-1/2 J +3/2 J +1 J J J -1/2 x y M xx M
IEEE Transactions on Antennas and Propagation, 2000
To reduce numerical dispersion in finite-difference time-domain (FDTD) methods, large computational stencils are often used. This paper proposes an optimized two-dimensional method by weighting the (2,4) stencil and the "neighborhood" stencil. After obtaining the amplification factor and the numerical dispersion relation, the optimal value of the weight parameter is obtained to minimize the numerical dispersion at a designated frequency. The anisotropy, dispersion error and the accumulated phase errors are greatly reduced over a broad bandwidth. Both the maximum anisotropy and the maximum dispersion error are 8.9
Coarse-Grid Higher Order Finite-Difference Time-Domain Algorithm With Low Dispersion Errors
… , IEEE Transactions on, 2008
Higher order (2,4) FDTD schemes used for numerical solutions of Maxwell's equations are focused on diminishing the truncation errors caused by the Taylor series expansion of the spatial derivatives. These schemes use a larger computational stencil, which generally makes use of the two constant coefficients, 1 and 2 , for the four-point central-difference operators. In this paper we propose a novel way to diminish these truncation errors, in order to obtain more accurate numerical solutions of Maxwell's equations. For such purpose, we present a method to individually optimize the pair of coefficients, 1 and 2 , based on any desired grid size resolution and size of time step. Particularly, we are interested in using coarser grid discretizations to be able to simulate electrically large domains. The results of our optimization algorithm show a significant reduction in dispersion error and numerical anisotropy for all modeled grid size resolutions. Numerical simulations of free-space propagation verifies the very promising theoretical results. The model is also shown to perform well in more complex, realistic scenarios.
A Novel Low-Dispersive (2, 2) Finite Difference Method: 3-D Case
ursi.org
The dispersion relation and convergence of a novel (2,2) finite-difference modified [5]-[8] (FDM) scheme which has fourth order convergence and excellent broadband characteristics, are presented. Accuracy of several low-dispersion finite-difference time-domain (FDTD) schemes in 2-D is compared with that of the FDM, via direct evaluation of the dispersion relation. Convergence of the FDTD and the FDM in 2-D are also examined.
A novel low-dispersive [2,2] finite difference method
Workshop on Computational Electromagnetics in Time-Domain, 2005. CEM-TD 2005., 2005
The dispersion relation and convergence of a novel (2,2) modified finite-difference time-domain (MFDTD) method with fourth order convergence and excellent broadband characteristics is presented. The accuracy of MFDTD is compared with that of standard FDTD and Fang (4,4) FDTD. The convergence characteristics of the MFDTD and the FDTD are also furnished. We have presented MFDTD in 2-D [1]. Here we extend MFDTD to the 3-D case.
An FDTD Algorithm for Wave Propagation in Dispersive Media using Higher-Order Schemes
Journal of Electromagnetic Waves and Applications, 2004
A fourth-order accurate in space and second-order accurate in time, Finite-Difference Time-Domain (FDTD) scheme for wave propagation in lossy dispersive media is presented. The formulation of Maxwell's equations is fully described and an elaborate study of the stability and dispersion properties of the resulting algorithm is conducted. The efficiency of the proposed FDTD(2,4) technique compared to its conventional FDTD(2,2) counterpart is demonstrated through numerical results.
Computers in Physics, 1994
We introduce a new second-order finite-difference time-domain (FDTD) algorithm to solve the wave equation on a coarse grid with a solution error less than 10−4 that of the conventional one. Although the computational load per time step is greater, it is more than offset by a large reduction in the number of grid points needed, while maintaining high accuracy, as well as by a reduction in the number of iterations. In addition, boundaries can be more accurately characterized at the subgrid level. This algorithm is based on a second-order finite-difference Laplacian that is nearly isotropic with respect to the wave propagation direction. Although optimum performance is achieved at a fixed frequency, the accuracy is still much higher than that of a conventional FDTD algorithm over ‘‘moderate’’ bandwidths.
IEEE Transactions on Antennas and Propagation, 2000
The numerical dispersion relations of finite-difference time-domain (FDTD) methods have been analyzed extensively in lossless media. This paper investigates numerical dispersion and loss for Yee's FDTD in lossy media. It is shown that: the numerical velocity can be smaller or larger than the physical velocity; there is no "magic time step size" in lossy media; and the numerical loss is smallest at the Courant limit. It is shown that the numerical loss is always larger than its physical value, and so Yee's FDTD overestimates the absorption of electromagnetic energy in lossy media. The numerical velocity anisotropy can be positive or negative, but the numerical loss anisotropy is always positive. The anisotropies in the three-dimensional (3-D) case are usually larger than those in the 2-D case. Numerical experiments in 1-D are shown to agree with the theoretical prediction.
A fourth-order accurate finite-difference scheme for the computation of elastic waves
1986
We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (µ(x)u x ) x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (µ(x)u x ) x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.