## 4 Numerical Examples
### Numerical Example 4  
The governing equation of the numerical example 4 is given as Equation (1). The initial condition is given as Equation (24). The left tidal boundary head is given as  
\[h(x=0,t)=A\sin(\frac{2\pi}{t_{0}}t). \tag{30}\]  
The boundary head on the inland side is described as  
\[\frac{\partial h(x=L_{f},t)}{\partial n}=0. \tag{31}\]  
The analytical solution [36] is found as  
\[h(x,t)=Ae^{-\alpha x}\sin(\frac{2\pi}{t_{0}}t-\frac{\pi S}{\alpha Tt_{0}}x), \tag{32}\]  
\[\alpha=\frac{1}{\sqrt{2}}\left\{\left[\left(\frac{L}{T}\right)^{2}+\left( \frac{2\pi S}{Tt_{0}}\right)^{2}\right]^{1/2}+\frac{L}{T}\right\}^{1/2}. \tag{33}\]  
Figure 8a depicts the schematic illustration of the one-dimensional tide-induced groundwater problem. The dimension of the length is 3000 m. The transmissivity is 2000 m\({}^{2}\)/d. The storage coefficient is \(10^{-3}\). The amplitude of the tidal change is 0.65 m. The tidal period is 24 hr. The leakage coefficient is considered to be 0/d. The total elapsed time is considered to be 3 hr. For the SCA analysis, we consider 183 boundary points and one source point. We consider the basis function order for the analysis to be 15. We also collocate 3721 inner points for the validation. Figure 11 indicates the computed and exact solutions [36]. From Figure 11, the computed tide-induced groundwater fluctuations are consistent with the exact solution. Using the SCA, the MAE is of the order of \(10^{-6}\). It is obvious that the SCA can acquire high accuracy performance. Figure 10: Result comparison of example 3 (case B). To further illustrate the correctness of the computed tide-induced head fluctuations from the proposed SCA and that from the conventional time-marching FDM, we conduct a comparison example in numerical example 4. The governing equation is represented by Equation (1). The initial condition is given as Equation (24). The left tidal boundary head is given as  
\[h(x=0,t)=A\cos(\frac{2\pi}{t_{0}}t). \tag{34}\]  
The boundary data on the inland side is described as  
\[\frac{\partial h(x=L_{f},t)}{\partial n}=0. \tag{35}\]  
The analytical solution is obtained as follows. \[h(x,t)=Ae^{-\alpha x}\cos(\frac{2\pi}{t_{0}}t-\frac{\pi S}{\alpha Tt_{0}}x), \tag{36}\]  
\[\alpha=\frac{1}{\sqrt{2}}\left(\left[\left(\frac{L}{T}\right)^{2}+\left(\frac {2\pi S}{Tt_{0}}\right)^{2}\right]^{1/2}+\frac{L}{T}\right)^{1/2}. \tag{37}\]  
Figure 8a depicts the schematic illustration of the one-dimensional tide-induced groundwater problem. The dimension of the length is 3000 m. The transmissivity is 2000 m\({}^{2}\)/d. The storage coefficient is \(10^{-3}\). The amplitude of the tidal change is 0.65 m. The tidal period is 24 h. The leakage coefficient is considered to be 0, 0.01, and 0.05/d. Therefore, the amplitude of groundwater head is calculated for different leakage coefficient. The total elapsed time is considered to be 3 and 6 hr. This example is solved utilizing both FDM and SCA. For the FDM analysis, the implicit scheme and the central difference approximation are considered for the temporal and spatial discretization, respectively. For the SCA analysis, we consider 183 boundary points and one source point. We consider the basis function order for the analysis to be 15. To obtain the field solution, we collocate 3721 inner points. Figures 12 and 13 depict the computed results from the analytical solution [13], the Laplace domain solution [37], the FDM, and our method at \(tf=3\) hr and \(tf=6\) hr, respectively. It is found that the computed results of tide-induced groundwater fluctuations using our method closely agree with the analytical solution. Figure 14 demonstrates the comparison of the accuracy with the FDM and the proposed method \(tf=3\) hr with the consideration of different leakage coefficient. We, therefore, compare the absolute error of our approach and those of the FDM. From Figure 14, the MAE of the  
Figure 11: Comparison of the computed head fluctuation with the analytical solution.