## 4 Numerical Examples
### Numerical Example 1  
Since our numerical method is established from the CTM, the correctness of the numerical solutions may depend on the terms of the Trefftz functions. Hence, a sensitivity study is conducted. The governing equation of one-dimensional tide-induced groundwater response in a confined aquifer is described as Equation (1). The data of the boundary are assigned by adopting the following analytical solution.  
\[h(x,t)=e^{-(\frac{L+T}{S})t}\sin x-e^{-(\frac{3t}{S})t}\cos(\sqrt{\frac{2L}{T}}x). \tag{14}\]  
The initial data is  
\[h(x,t=0)=-\cos(\sqrt{\frac{2L}{T}}x)+\sin x. \tag{15}\]  
Figure 2a depicts the configuration of the one-dimensional tide-induced groundwater problem. The space dimension of the problem (\(L_{f}\)) is 1000 m. The transmissivity (\(T\)) is 1800 m\({}^{2}\)/hr. The storage coefficient (\(S\)) is \(2\times 10^{-4}\). The leakage coefficient (\(L\)) is 0.12/d. The elapsed time (\(t_{f}\)) is 120 hr. The numbers of the boundary and source points in this example are considered to be 150 and one. The initial data are provided on the bottom spacetime region. As for the boundary data, it is assigned on both right and left of the spacetime region, as illustrated in Figure 2. To evaluate the accuracy of the proposed approach, the maximum absolute error (MAE) is evaluated by the following equation [32; 33; 34].  
\[\text{MAE}=\text{max}|h_{e}-h_{n}|, \tag{16}\]  
where \(h_{e}\) denotes the exact solution and \(h_{n}\) denotes the numerical solution. Figure 3 demonstrates the terms of the basis function versus the MAE. It seems that the computed results with high accuracy may be obtained when the terms of the basis function is greater than 10. To further investigate the accuracy of the SCA, we consider another problem. The governing equation can be described as Equation (1). The initial condition is described as  
\[h(x,t=0)=x. \tag{17}\]  
Figure 3: The basis function order versus the maximum absolute error (MAE).  
The right, as well as left boundary conditions are described as  
\[h(x=L_{f},t)=xe^{-\frac{L}{5}t}, \tag{18}\]  
\[h(x=0,t)=0. \tag{19}\]  
The analytical solution is given as  
\[h(x,t)=xe^{-\frac{L}{5}t}. \tag{20}\]  
The space dimension of the problem is 5000 m. The transmissivity is 1.25 m\({}^{2}\)/hr. The storage coefficient is \(2\times 10^{-4}\). The leakage coefficient is \(5\times 10^{-3}\)/d. The elapsed time is 5 hr. The schematic illustration is shown in Figure 2a. The initial and boundary values are provided on the lateral and bottom of the spacetime region, as displayed in Figure 2b. There exist one source point and 153 boundary points. We consider the terms of the basis function to be 10. To yield the field solution, we consider 2601 inner points distributed in the spacetime domain. Figure 4 depicts the computed results with the exact solution. It seems that the computed groundwater head fluctuation using the SCA may closely agree with the analytical solution. Figure 5 shows the correctness for the SCA. The MAE associated with the SCA is of the order of \(8.9\times 10^{-16}\). It is obvious that the SCA may acquire highly accurate results.  
Figure 4: Result comparison of example 1.