A Novel Gram-Schmidt-Based Optimal Finite Difference Method for Acoustic Wave Modeling

As a common technique in seismic-wave numerical simulation, the finite difference method inevitably encounters issues when it comes to numerical dispersion. Therefore, various optimization methods were proposed to suppress this problem. Reversal of the trigonometric function's Taylor expansion is achieved through Gram-Schmidt orthogonalization, where the expanded equation is converted back to trigonometric form to minimize numerical dispersion. The Gram–Schmidt procedure method is a simple linear algebraic algorithm converting original vectors via an orthonormal approach. The proposed methodology additionally incorporates the binomial theorem in conjunction with Euler's formula, thereby enhancing computational efficiency. To demonstrate ability of the Gram–Schmidt procedure method to suppress numerical dispersion, we compared it with that of the traditional Taylor expansion and novel least squares methods. Compared with that of the traditional comparison within a certain wavenumber range, we conduct comparative analysis through a full wavenumber range (small, medium, and wide wavenumber ranges), for more accurate comparison results. First, the error curves of the three methods were compared using the space-domain dispersion relation equation. Second, simple uniform and complex models were used for the comparative test. The experimental results demonstrate that the Gram-Schmidt orthogonalization-based approach exhibits superior performance compared to both Taylor expansion and least squares methods.