Scalar wave equation modeling with dispersion relation based on finite difference method

The finite-difference method(FDM) is widely used in the numerical modeling of wave equations. Conventional FDM stencils for spatial derivatives are usually designed in the space domain, which creates difficulty in satisfying the dispersion relations exactly while solving the wave equations. We use an automated and optimized FDM using a genetic algorithm to optimally compute second-order spatial derivatives. In our method, the explicit finite-difference stencils are calculated using the genetic algorithm to minimize the dispersion (phase velocity) for all wavenumbers without using any specific window function. The amplitudes of the pseudo-spectral window are optimized by making the phase velocity close to the analytical solution at each wavenumber, where the stability is close to that of the conventional FDM. Although finite difference coefficients in this method depend on velocity, grid spacing and time step, less dispersive solutions can be achieved by computing suitable finite-difference coefficients for varying cases. We compare our results with the solutions of an existing pseudo-spectral method (with Kaiser window function), conventional FDM, joint time-space optimization method, and the least square method. The normalized phase velocity and the absolute error of our method show very promising results.