## 3 Calculation Simulation and Analysis
### Performance Comparison of Different Methods
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **EEM** & **EIM** & **IIM** & **GEM** & **GIM** \\ \hline time/s & 243 & 249 & 711 & 445 & 447 \\ \(p_{1}\) & 0.0208 & 0.0207 & 0.0160 & 0.0162 & 0.0160 \\ \(p_{2}\) & 0.0087 & 0.0082 & 0.0019 & 0.0047 & 0.0019 \\ \(u_{3}\) & 0.0019 & 0.0019 & 0.0001 & 0.0009 & 0.0001 \\ \(u_{4}\) & 0.0020 & 0.0020 & \(7.1\times 10^{-5}\) & 0.0010 & \(8.3\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Root mean square error at a step size of 0.0001 s.  
Figure 4: Calculated results for the first 100 time steps of Equation (20) (displacement at reception point 3). COMSOL simulations at a time step of 0.0001 s. It can be seen that the field variables at each receiving point calculated by the GIM method are in very good agreement with the \"comparison solution\". The root mean square error of the five methods when the time step is taken as 0.0003 s is shown in Table 2.  
Figure 5: GIM calculation results for a step size of 0.0001 s (sound pressure at reception point 1). Figure 8: GIM calculation results for a step size of 0.0001 s (displacement at reception point 3). Figure 6: GIM calculation results for a step size of 0.0001 s (sound pressure at reception point 2). Figure 7: GIM calculation results for a step size of 0.0001 s (displacement at reception point 2). As can be seen from Table 2, since the step size is chosen close to the stability condition of the explicit method, the errors of EEM, EIM, and GEM with the explicit method increase, while the errors of the unconditional implicit methods IIM and GIM are still close to the \"comparative solution\". When the time step is 0.0004 s, the explicit method is unstable, the EEM and GEM do not satisfy the stability condition of Equation (25), and the results do not converge. Due to the mixing of implicit algorithms in EIM, the results still converge, and their root mean square errors with IIM and GIM are shown in Table 3.  
As can be seen from Table 3, the computational errors of all three methods increase, but a comparison with the EEM in Table 1 shows that the computational accuracy of GIM at a time step of 0.0004 s is better than that of the EEM at 0.0001 s, and the time consumed is about half of that of the EEM, which at this time can reflect the advantage of the global implicit solving method at larger step sizes. When the time step is taken as 0.001 s, only IIM and GIM converge, and their root mean square errors are shown in Table 4.  
As can be seen from Table 4, the field variable errors at the coupling boundary are significantly larger for GIM compared to GIM with a step size of 0.0001 s. The acoustic pressure error in the fluid increases by 0.02, and the displacement error in the elastomer increases by about \(10^{-3}\) orders of magnitude. The increase in the field variable errors in the fluid and the elastomer for GIM compared to EEM with a step size of 0.0001 s is within 0.01, while the elapsed time is reduced by one-fifth. In summary, for the time-domain solution of the shallow water seismic wave field equations, the implicit method is superior to the explicit method in terms of stability because the stability of the explicit method is limited by the time-step requirement in Equation (25), while the implicit method is unconditionally stable; the explicit method is superior to the implicit method in terms of computational efficiency because the explicit method requires only one equation to iterate for each of the field variables to be solved, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **IIM** & **GIM** \\ \hline time/s & 71 & 44 \\ \(p_{1}\) & 0.0344 & 0.0344 \\ \(p_{2}\) & 0.0571 & 0.0571 \\ \(u_{3}\) & 0.0113 & 0.0113 \\ \(u_{4}\) & 0.0090 & 0.0090 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Root mean square error at a step size of 0.001 s.