Second-order implicit finite-difference schemes for the acoustic wave equation in the time-space domain

Abstract

We have developed compact implicit finite-difference (FD) schemes in the time-space domain based on the second-order FD approximation for accurate solution of the acoustic wave equation in 1D, 2D, and 3D. Our method is based on the weighted linear combination of the second-order FD operators with different spatial orientations to mitigate numerical error anisotropy and the weighted averaging of the mass acceleration term over the grid points of the second-order FD stencil to reduce the overall numerical dispersion error. We have developed a derivation of the schemes for 1D, 2D, and 3D cases. We obtain their corresponding dispersion equations, then we find the optimum weights by optimization of the time-space domain dispersion function, and finally we tabulate the optimized weights for each case. We analyze the numerical dispersion, stability, and convergence rates of our schemes and compare their numerical dispersion characteristics with the standard high-order ones. We also discuss the efficient solution of the system of equations associated with our implicit schemes using the conjugate-gradient method. The comparison of dispersion curves and the numerical solutions with the analytical and the pseudospectral solutions reveals that our schemes have better performance than the standard spatial high-order schemes and remain stable for relatively large time steps.