We develop a novel asymptotic local finite-difference (ALFD) method for solving the fractional Laplacian wave equation in the time domain. This numerical method uses the trigonometric function-based generating function to derive the quasidifference operator, facilitating the discretization of the fractional Laplacian without requiring fast Fourier transforms during simulations. Theoretical analysis and numerical experiments demonstrate that the ALFD method is accurate and efficient. Stability analysis using the eigenvalue method reveals that our method enjoys a relaxed stability condition (the minimum Courant number of 0.659492), providing greater flexibility in the selection of sampling intervals. Numerical dispersion anisotropy is significantly mitigated for various very small Q values (Q < 10). The numerical solutions obtained via the ALFD method march closely with the analytical solutions derived from Green’s function. The computation speed of ALFD is significantly higher than the analytical method and the original global finite-difference (GFD) method. For a 2D model with 400 × 400 grid points, the computational time required by the ALFD method is approximately 9.96%, 5.40%, 4.94%, and 4.60% of that required by the analytical method, and only 2.54%, 1.37%, 1.25%, and 1.15% of that required by the GFD method, respectively. Numerical examples illustrate the effectiveness of the ALFD method in simulating heterogeneous attenuating media, highlighting its potential as a valuable tool for studying inverse problems in viscoelastic media.

You do not have access to this content, please speak to your institutional administrator if you feel you should have access.