High-Accuracy Finite-Difference Schemes for Linear Wave Propagation

Abstract

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High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

“…Thus, the IDM yields an excellent way to generalize the embedding FDTD scheme to high orders. Furthermore, it could be applied together with various different high-order time-domain methods, such as high-order FDTD methods [3,[10][11][12][13][14][15][16], the MRTD method [17,18], and the LSTD method [24][25][26]. In these applications, the number of FPs (m) could be simply specified according to the length of stencil involved in these time-domain methods, and significant improvement of accuracy is expected.…”

“…For broadband applications and problems including material interface, wave scattering and penetration over large and complex domains, the grid size required by using the FDTD method could become prohibitively expensive for modern computers. Much progress has been made in the past two decades in improving the FDTD method, including plentiful methods for removing the staircased approximation for boundaries and geometries, [5][6][7][8][9] and numerous high-order FDTD methods [3,[10][11][12][13][14][15][16]. Here, by high order we refer to orders being higher than three, which are essential for modern problems of moderately high frequency (short) waves and/or large domain in nature.…”

Coalition Logistics: The Way to Win the Peace, The Way to Win the War

“…Because of the difference in group velocity for different wavenumbers, the wavepacket will disperse in time. and Zingg et al (1996Zingg et al ( , 2000 based the analysis of numerical dispersion on phase velocity errors of a finite-difference discretization when, in fact, dispersion errors of a dispersive system are determined by the group velocity (Whitham, 1974). explicit optimized (Zingg, 1996) − ⋅ ⋅ − ⋅ ⋅ −; optimized compact (Lele, 1992); − − − −; optimized compact (Haras, 1994) − ⋅ − ⋅ −; prefactored 6 th -order compact (Hixon, 1998a) ⎯•⎯ ; optimized upwind ………; optimized upwind for the seven schemes: DRP (Tam, 1993a), explicit optimized (Zingg, 1996), pentadiagonal "spectral-like" compact (Lele, 1992), tridiagonal optimized compact (Haras, 1994), prefactored 6 th -order compact (Hixon, 1998a), explicit optimized upwind , explicit high-bandwidth upwind .…”

“…The optimized operator of Zingg et al (1993Zingg et al ( , 1996 has also a seven-point stencil. The discretization is divided into a central-difference part approximating the derivative and a symmetric part providing artificial dissipation of spurious numerical waves, The coefficients a j and d j are calculated so that the approximation is formally first-order accurate and the four remaining coefficients are used as optimization parameters to minimize the maximum phase and amplitude errors for waves resolved with at least ten points per wavelength.…”

The spectral-element method in seismology

“…Unfortunately, this classical method suffers from limitations when addressing the complexity of typical 3-D models, such as the presence of surface topography or major discontinuities within the model [e.g., Robertsson, 1996;Ohminato and Chouet, 1997]. Recently developed optimal or compact finite-difference operators have improved this situation [e.g., Zingg et al, 1996;Zingg, 2000]. Methods that resort to more accurate spatial derivative operators, such as spectral and pseudospectral techniques based on global gridding of the model, have also been used to address regional [e.g., Carcione, 1994] and global [e.g., Tessmer et al, 1992] seismic wave propagation problems.…”