A central difference method with low numerical dispersion for solving the scalar wave equation

Abstract

A B S T R A C TIn this paper, we propose a nearly-analytic central difference method, which is an improved version of the central difference method. The new method is fourth-order accurate with respect to both space and time but uses only three grid points in spatial directions. The stability criteria and numerical dispersion for the new scheme are analysed in detail. We also apply the nearly-analytic central difference method to 1D and 2D cases to compute synthetic seismograms. For comparison, the fourthorder Lax-Wendroff correction scheme and the fourth-order staggered-grid finitedifference method are used to model acoustic wavefields. Numerical results indicate that the nearly-analytic central difference method can be used to solve large-scale problems because it effectively suppresses numerical dispersion caused by discretizing the scalar wave equation when too coarse grids are used. Meanwhile, numerical results show that the minimum sampling rate of the nearly-analytic central difference method is about 2.5 points per minimal wavelength for eliminating numerical dispersion, resulting that the nearly-analytic central difference method can save greatly both computational costs and storage space as contrasted to other high-order finitedifference methods such as the fourth-order Lax-Wendroff correction scheme and the fourth-order staggered-grid finite-difference method. I N T R O D U C T I O NDuring the past several decades, a great increase in computer performance and saving memory have driven wide applications of numerical methods in acoustic phenomena in complex media. The elastic properties of complex media with fractures, cracks or pores show significant spatial variations. Any hope of analysing synthetic seismograms relies heavily on the accurate resolution of a wide range of temporal and spatial scales present in acoustics and seismology. Hence, it is essential to develop efficient numerical methods with high accuracy and fast computational speed. *

Wave-equation-based travel-time seismic tomography – Part 1: Method

“…Appendix A: High-order central difference method Yang et al (2012) developed a finite-difference scheme, nearly analytic central difference (NACD) method to solve the 2-D acoustic wave equation. The NACD method has fourth-order accuracies in both space and time, and it uses only three grid nodes in each spatial direction.…”

Evaluation of dispersion characteristics of multimodal guided waves using slant stack transform

“…The discrepancy increases with frequency, which can be explained by the numerical errors that increase when the ratio of the wavelength to the cell dimension used in the simulations decreases. Other discrepancies result from numerical dispersion introduced by the numerical discrete time and space LISA model [22,23].…”

3D weak-dispersion reverse time migration using a stereo-modeling operator

“…The discretization of fourthand fifth-order derivatives with respect to x, y, and z are the same as those in the previous work (Yang et al, 2007). Then, following the directional derivatives approach, which was introduced by Yang et al (2012), we get the stereo-modeling type expressions for all the mixed spatial partial derivatives needed, which are different from the previous work (Yang et al, 2007(Yang et al, , 2012. For convenience, we list the expressions of all the derivatives needed in computation in Appendix A.…”

“…Then, following the methodology that has been previously described for the 2D case of NACD (Yang et al, 2012), we apply the central difference discretization to the second-order spatial derivatives and keep the high-order terms required to achieve fourth-order accuracy in space. This can be seen as doing Taylor expansions in space.…”