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

Abstract

Abstract The staggered grid finite difference method is widely used in the 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 mainly converts the acoustic equation into a polynomial equation of the trigonometric function and then 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 view of this, we used the Gram–Schmidt orthogonalization method, combined with the binomial theorem and Euler formula, to reverse the polynomial of power function into a polynomial of trigonometric function and finally obtain a new difference coefficient. To highlight the effectiveness of our new method, we compared it with the Taylor expansion and least-squares methods by selecting a small wavenumber, middle wavenumber, and wide wavenumber ranges. First, accuracy and dispersion analyses were conducted, and the results showed that the new difference coefficient generated smaller errors and induced stronger suppression of the numerical dispersion. We conducted a comparative analysis of the uniform and complex models, which further validated the superiority of the proposed staggered grid difference coefficient.