Optimal Nearly Analytic Discrete Approximation to the Scalar Wave Equation

A Fourth-Order Runge-Kutta Method with Low Numerical Dispersion for Simulating 3D Wave Propagation

“…Following the Fourier analysis (Richtmyer & Morton, 1967;Yang et al, 2006Yang et al, , 2010, after some mathematical derivations (see Appendix B for detail), we obtain the stability conditions for solving 1D, 2D, and 3D acoustic equation as follows:…”

“…But these high-order schemes also can not cure the numerical dispersion effectively when coarse grids are used, and they usually involve in more grids in a spatial direction than low-order schemes (Yang et al, 2006). For example, the tenth-order compact FD scheme (e.g., Wang et al, 2002), which usually uses more grids than low order schemes, also suffers from numerical dispersion.…”

“…This implies that two kinds of dispersions (numerical dispersion and physical dispersion) might occur simultaneously in wave fields if the conventional FD methods are used to compute the wave fields in a porous medium. In such a case, it is not a good idea to use the FCT technique to eliminate the numerical dispersions because we do not know how to choose the proper control parameters used in the FCT method for suppressing the numerical dispersions (Yang et al, 2006). The pseudo-spectral method (PSM) is attractive as the space operators are exact up to the Nyquist frequency, but it requires the Fourier transform (FFT) of wave-field to be made, which is computationally expensive for 3D anisotropic models and has the difficulties of handling non-periodic boundary conditions and the non-locality on memory access of the FFT, which makes the parallel implementation of the algorithms and boundary treatments less efficient (Mizutani et al, 2000).…”

“…Meanwhile, it also suffers from numerical dispersion in the time direction, and its numerical dispersion is serious as the Courant number, defined by 0 / ct x    (Dablain 1986;Sei & Symes, 1995), is large, i.e. as the time increment is large (Yang et al, 2006). Another easy way to deal with the numerical dispersion is to use fine grids to increase spatial samples per wavelength.…”

“…Fortunately, with the rapid development of computer performance and the birth of parallel technology in past several decades, 3D wave simulation through using different numerical methods on a large scale or high frequencies becomes affordable, and the study of 3D numerical techniques has been a hot spot and rapidly developed because of its applying to practical issues in the fields of Acoustics and Geophysics. Recently, the so-called nearly analytic discrete (NAD) method and optimal NAD (ONAD) (Yang et al, 2006) suggested by Yang et al (2003) for acoustic and elastic equations, which was initially developed by Konddoh et al (1994) for solving parabolic and hyperbolic www.intechopen.com A Fourth-Order Runge-Kutta Method with Low Numerical Dispersion for Simulating 3D Wave Propagation 213 equations, is another effective method for decreasing the numerical dispersion. The method, based on the truncated Taylor expansion and the local interpolation compensation for the truncated Taylor series, uses the wave displacement-, velocity-and their gradient fields to restructure the wave displacement-fields.…”