Acoustic wave propagation with new spatial implicit and temporal high-order staggered-grid finite-difference schemes

Abstract

Abstract The implicit staggered-grid (SG) finite-difference (FD) method can obtain significant improvement in spatial accuracy for performing numerical simulations of wave equations. Normally, the second-order central grid FD formulas are used to approximate the temporal derivatives, and a relatively fine time step has to be used to reduce the temporal dispersion. To obtain high accuracy both in space and time, we propose a new spatial implicit and temporal high-order SG FD stencil in the time–space domain by incorporating some additional grid points to the conventional implicit FD one. Instead of attaining the implicit FD coefficients by approximating spatial derivatives only, we calculate the coefficients by approximating the temporal and spatial derivatives simultaneously through matching the dispersion formula of the seismic wave equation and compute the FD coefficients of our new stencil by two schemes. The first one is adopting a variable substitution-based Taylor-series expansion (TE) to derive the FD coefficients, which can attain (2M + 2)th-order spatial accuracy and (2N)th-order temporal accuracy. Note that the dispersion formula of our new stencil is non-linear with respect to the axial and off-axial FD coefficients, it is complicated to obtain the optimal spatial and temporal FD coefficients simultaneously. To tackle the issue, we further develop a linear optimisation strategy by minimising the L2-norm errors of the dispersion formula to further improve the accuracy. Dispersion analysis, stability analysis and modelling examples demonstrate the accuracy, stability and efficiency advantages of our two new schemes.