High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation

Finite-difference (FD) methods are widely used for numerical solution of acoustic and elastic wave equations. Temporal high-order FD methods exhibit better accuracy and stability than the methods with second-order differencing in time. Also, the implicit calculation of spatial derivatives can bring significant improvement in accuracy. The present implicit FD methods with high-order accuracy in time are based on centred grids. In this paper, we propose an implicit staggered-grid FD (SFD) scheme with a combined stencil, which is the combination of rhombus/pyramid and cross stencils in 2-D/3-D case, for modelling scalar wave propagation. Our scheme computes the temporal and spatial derivatives using high-order temporal and implicit spatial FD operators based on the combined stencil and the conventional stencil, respectively. We derive the dispersion relations of the FD scheme for 2-D and 3-D scalar wave equations and estimate temporal and implicit spatial FD coefficients by Taylor series expansion (TE) and least squares (LS). According to the kinds of FD coefficients, we formulate four implicit SFD operators: TE-TE, TE-LS, LS-TE and LS-LS operators. We carry out the comparison between our scheme and several existing SFD schemes: the conventional explicit and implicit, optimal explicit and implicit and explicit temporal high-order schemes. 2-D and 3-D dispersion analysis, stability analysis and modelling examples reveal that our implicit scheme has greater accuracy than other schemes and requires slightly stricter stability condition than the explicit temporal high-order SFD schemes. Owing to higher accuracy, our implicit SFD scheme with LS-LS operators allows for shorter FD operators and larger grid spacing, which can increase the computational efficiency.