*3.5. Time Dispersion Suppression Comparison*

Seismic waves propagate in time and space. The numerical dispersion caused by grid discretization includes both spatial dispersion and time dispersion. The accuracy of time extrapolation of the traditional high-order difference (2 m, 2) scheme is only second order. When a large time step is adopted, there will be obvious time dispersion.

We use the same model to evaluate time dispersion as used for spatial dispersion. To evaluate time dispersion, it is necessary to eliminate the influence of numerical dispersion caused by the spatial gird size. Therefore, the spatial grid size is set to 5 m, and we increase the time step to 4 ms. The calculated seismic records of CCD and CFD are shown in Figure 6a,b.

**Figure 6.** (**a**) CFD is used to simulate the numerical simulation of seismic records in the 1D homogeneous medium model, Δt = 0.004 s; (**b**) CCD is used to simulate the numerical simulation of seismic records in the 1D homogeneous medium model, Δt = 0.004 s.

The seismic record simulated using the CFD scheme shows serious and obvious dispersion, while the result of the CCD scheme shows almost no such dispersion. As the CCD scheme proposed in this paper uses the fourth-order difference operator to approximate the time partial derivative, it has better stability than the CFD scheme, which can only use the second-order difference operator to approximate the time partial derivative. The CSCD scheme, by contrast, uses the same fourth-order difference operator to approximate the time partial derivative, which shows even higher accuracy than the CCD scheme.

Figure 7a,b compare the seismic records using the fourth- and second-order difference operators to approximate the time partial derivative from the CCD scheme. It can be seen that when using the same finite difference scheme to suppress the numerical dispersion caused by the spatial gird size, the fourth-order operators show smaller numerical dispersion caused by increasing time step size than the second-order operators, which provides a theoretical basis for the use of large spatial gird size and large time step size for shear-wave simulation.

**Figure 7.** (**a**) CCD is used to simulate the numerical simulation of seismic records in the 1D homogeneous medium model with fourth-order accuracy in time, dt = 0.004 s; (**b**) CCD is used to simulate the numerical simulation of seismic records in the 1D homogeneous medium model with second-order accuracy in time, dt = 0.004 s.
