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.