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.