An optimized finite-difference method to minimize numerical dispersion of acoustic wave propagation using a genetic algorithm

Abstract

The finite-difference method (FDM) is one of the most popular methods for numerical simulation of wave propagation. The major challenge that we face in solving a partial differential equation for the wave propagation is numerical dispersion. We use an automated and optimized FDM using a genetic algorithm (GA) to optimally compute second-order spatial derivatives. In our method, the explicit finite-difference coefficients are calculated using the GA to minimize the dispersion (phase velocity) for all wavenumbers without using any specific window function. The amplitudes of the pseudospectral window (finite difference [FD] coefficients) are optimized by making the phase velocity close to the analytical solution at each wavenumber and stability close to that of the conventional FDM. Although FD coefficients in this method depend on velocity, grid spacing, and time step, less dispersive solutions can be achieved by computing suitable FD coefficients for varying cases. It takes only a few seconds of additional computation time depending on the order of approximation (e.g., approximately 20 s for the 12th order of approximation in a machine with 120 GB RAM and 2.0 GHz processor). Comparison of the numerical results from our algorithm with those from an existing pseudospectral method, the conventional FDM, and an existing time-space optimized FDM indicates that the proposed method is less dispersive for any order of approximation.