**3. Analysis of Combined Compact Difference Scheme**

In numerical simulations, if the size of the spatial grid is too large, it will cause large solution errors and produce numerical dispersion [37,38]. The simple harmonic wave solution is introduced to have the velocity ratio curve at different *θ* values in the isotropic case; *θ* is the angle between the wave's propagation direction and the *x*-axis. It is used to compare the spatial dispersion suppression effects of CCD with the traditional finite difference scheme. It is shown as follows:

In Figure 2, the ratio of numerical wave velocity to true velocity is defined as:

$$
\lambda = \frac{v\_{num}}{v} \tag{4}
$$

where *vnum* is the numerical wave velocity, and *v* is the true wave velocity. It is shown in Figure 2 that with the velocity ratio curves of the CCD, the other two different formats at different *θ* values. The Courant numbers (*α* = *v*Δ*t*/*h*) are 0.25, the horizontal axis *ϕ* ∈ [0, *π*] is the product of the wavenumber and the spatial step, and the y-coordinate is the velocity ratio *λ*, with 1 meaning that the numerical wave speed is consistent with the theoretical wave speed and indicating that the numerical dispersion is low. It also indicates that CCD has the best suppression effects.

**Figure 2.** Velocity ratio curves for different numerical simulation methods. The blue curve is the traditional central difference scheme, the green curve is the traditional implicit difference scheme, the red curve is the CCD difference scheme and the black line is the velocity ratio constant of 1: (**a**) *θ* = 0, *α* = 0.25; (**b**) *θ* = *π*/6, *α* = 0.25; (**c**) *θ* = *π*/3, *α* = 0.25.

Additionally, the simulation error is calculated by simulating the two-dimensional plane harmonic initial value problem to analyze and compare the numerical simulation accuracy of the CCD and CFD (centered finite difference scheme).

In Figure 3, the simulation results show that the accuracy is relatively high for the SH shear wave simulation results with the adoption of the CCD format, and the numerical simulation of the seismic wave field with a large sampling time can be performed [39].

**Figure 3.** Relative errors of numerical simulation for different schemes and gird size: (**a**) Δx = 10 m, Δt = 0.001 s; (**b**) Δx = 15 m, Δt = 0.001 s.
