Novel Optimal Staggered Grid Finite Difference Scheme Based on Gram–Schmidt Procedure for Acoustic Wave Modelling

Abstract

The staggered grid finite difference method is used extensively in numerical simulations of acoustic equations; however, its application is accompanied by numerical dispersion. The most representative traditional method for suppressing the numerical dispersion is the Taylor expansion method, which primarily converts the acoustic equation into a polynomial equation of the trigonometric function and subsequently expands the trigonometric function into a power function polynomial through the Taylor expansion to finally obtain the difference coefficient. However, this traditional method is only applicable to the small wavenumber range. In this study, the Gram–Schmidt orthogonalization method, combined with the binomial theorem and Euler formula was used to reverse the polynomial of power function into a polynomial of trigonometric function and to obtain a new difference coefficient. To highlight the effectiveness of the proposed method, it was compared with the Taylor expansion method (TEM) and the least-squares method (LSM) by selecting a small wavenumber, middle wavenumber, and wide wavenumber ranges. Accuracy and dispersion analyses were conducted. The results showed that the new difference coefficient generated smaller errors and induced stronger suppression of the numerical dispersion. A comparative analysis of the uniform and complex models further validated the superiority of the proposed staggered grid difference coefficient.