4.4 Numerical example 4
To illustrate the accuracy of the computed result 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 described as Equation (1). The initial condition is described as Equation (31). The tidal boundary condition at left of the space domain is expressed as
. (37)
The boundary data on the inland side is described as
. (38)
The analytical solution is obtained as follows.
, (39)
. (40)
The dimension of the length is 3000 (m). The transmissivity is 2000 (m2/d). The storage coefficient is . The amplitude of the tidal change is 0.65 (m). The tidal period is 24 (hr). The leakage coefficient is considered to be 0, 0.01, and 0.05 (1/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. The order of the basis function for the analysis is 15. To obtain the field solution, we collocate 3721 inner points. Figure 11 and Figure 12 depict the computed results from the analytical solution (Jiao & Tang, 1999), the Laplace domain solution (Kim et al., 2003), the FDM, and our method at and , respectively. It is found that the computed results of tide–induced groundwater fluctuations using our method closely agree with the analytical solution. Figure 13 demonstrates the comparison of the accuracy with the FDM and the proposed method at with the consideration of different leakage coefficient. We therefore compare the absolute error of our approach and those of the FDM. From Figure 13, the MAE of the FDM is only on the order of 10-2 and 10-3. However, the MAE of our method may reach up to the order of 10-9, as displayed in Figure 13. Table 2 shows the accuracy of our method with those of the conventional time–stepping approach using the FDM with the consideration of different leakage coefficient. From Table 2, it is found that the MAE of our method remains in the order of 10-5 to 10-9. It is significant that excellent agreement is achieved and accurate numerical results can be obtained by the proposed method.