5. DISCUSSION
To investigate the proposed method, five numerical examples are
conducted. Based on the results, the discussions are carried out as
follows.
- We conduct the sensitivity analysis and results comparison with the
analytical solution in example 1 and example 2. Results obtained
demonstrate that the MAE can reach to the order of
10-16. It is significant that the SCA may obtain
highly accurate results. The accuracy of the proposed method is close
to that of the analytical method. To further demonstrate the
advantages of the proposed method, example 3 and 4 are conducted to
investigate the accuracy of temporal discretization. From results of
this study, we indicate that the MAE of the SCA remains in the order
of 10-5 to 10-9. We also compare
the MAE of our method and those of the FDM. The MAE of the FDM is only
on the order of 10-2 and 10-3. It
is significant that the proposed method may have more accurate
numerical results than those of the conventional time–marching FDM.
- In this article, we propose the spacetime collocation approach for
solving the tide–induced groundwater response. For the solving of the
tide–induced groundwater response in a coastal confined aquifer with
an estuary tidal–loading boundary using the proposed method, the
boundary points are placed in the spacetime region. Hence, both the
boundary and initial values are considered as boundary conditions on
the boundary of the spacetime. The one–dimensional initial value
problem can then be regarded as a two–dimensional inverse BVP.
Accordingly, the transient problems for tide–induced groundwater
response in a coastal confined aquifer may be solved without utilizing
the time–stepping techniques. Besides, more accurate numerical
results for solving transient problems can be obtained using the
proposed method. However, similar to other boundary–type meshless
methods, the proposed method may also suffer from the numerical
instability due to the ill–posed phenomenon of the meshless method
for solving two– or three–dimensional problems.