4.1 Numerical example 1
Since our numerical method is
established from the CTM, the accuracy of the numerical solutions for
the CTM may depend on the order of the Trefftz functions. Hence, a
sensitivity analysis of the order of the Trefftz basis functions 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.
. (22)
The initial data is
. (23)
The space dimension of the problem () is 1000 (m). The transmissivity
(T ) is 1800 (m2/hr). The storage coefficient
(S ) is . The leakage coefficient (L ) is 0.12 (1/d). The
elapsed time () 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 assigned on the
bottom of the spacetime region. As for the boundary data, it is assigned
on both sides of the spacetime region, as depicted in Figure 2. Figure 3
demonstrates the order of the basis function versus the maximum absolute
error (MAE). It seems that the computed results with high accuracy may
be obtained when the order of the basis function is greater than 10.
To further investigate the accuracy of the SCA, we consider another
problem. The governing equation is described as Equation (1). The
initial data is described as
. (24)
The boundary data at right and left of the space domain are expressed as
, (25)
. (26)
The analytical solution is obtained as
. (27)
The space dimension of the problem is 5000 (m). The transmissivity is
1.25 (m2/hr). The storage coefficient is . The leakage
coefficient is (1/d). The elapsed time is 5 (hr). The boundary and
initial values are assigned on the lateral and bottom sides of the
spacetime region, as displayed in Figure 2 (b).
There exist one source point and 153 boundary points. The order of the
basis function is 10. To yield the field solution, we consider 2601
inner points distributed inside the spacetime domain. The computed
results with the exact solution are illustrated in Figure 4. It seems
that the computed groundwater head fluctuation using the SCA may closely
agree with the analytical solution. The accuracy for the SCA is
illustrated in Figure 5. The MAE associated with the SCA can reach the
order of . It is significant that the SCA may obtain highly accurate
results in this example.
4.2 Numerical example 2
The governing equation of the numerical example 2 is described as
Equation (1). The boundary data at left of the space domain is described
as Equation (26). We consider the boundary data at right of the space
domain to be
. (28)
The initial condition is
. (29)
The exact solution is found as
. (30)
The dimension of the space domain, the transmissivity, the storage
coefficient, the leakage coefficient, and the elapsed time are assumed
to be 5000 (m), 1.25 (m2/hr), , (1/d) and 5 (hr),
respectively.
The boundary and initial data are given on the lateral and bottom sides
of the spacetime domain. We adopt 153 boundary points uniformly
distributed on the boundary and a source point. The Dirichlet and
Neumann boundary data are assigned on boundary points. The order of the
basis function is 10. To yield the field solution, we collocate 2601
inner points.
The computed results and the exact solution are illustrated in Figure 6.
From Figure 6, the results are entirely consistent with the exact
solution. The accuracy for the SCA is illustrated in Figure 7. Using the
SCA, the MAE reach up to the order of . It is obvious that the SCA can
obtain accurate results in this numerical example.