## 3 The Spacetime Collocation Approach  
The spacetime collocation scheme begins with the consideration of transient Trefftz basis functions which are also the general solutions of the governing equation. Thus, it may be necessary to derive general solutions for the tide-induced groundwater flow problem in a confined aquifer. To establish the transient Trefftz functions for this problem, the separation of variables is adopted by considering the transient solutions of the tide-induced groundwater flow problem. \[h(x,t)=X(x)K(t). \tag{5}\]  
For simplicity, the following equations are considered. \[X^{\prime\prime}=\frac{d^{2}X}{dx^{2}}\text{, and }K\prime=\frac{dK}{dt}. \tag{6}\]  
Inserting Equation (6) into Equation (1) can obtain the following equation. \[TX^{\prime\prime}K-LXK=SXK^{\prime}\text{ in }\Re_{t}. \tag{7}\]  
Dividing by \(X(x)K(t)\) in Equation (7), we may yield  
\[\frac{X^{\prime\prime}}{X}=\lambda,\text{ and }\frac{S}{T}\frac{K^{\prime}}{K} +\frac{L}{T}=\lambda, \tag{8}\]  
where \(\lambda\) is a separation constant. To evaluate the eigenvalue of Equation (8), the \(p\) constant is introduced to ensure this constant to be positive or negative value. Hence, we may consider \(\lambda=p^{2}\), \(\lambda=0\), and \(\lambda=-p^{2}\). The derivation of Trefftz functions for one-dimensional groundwater flow is listed in Appendix A. According to the superposition theorem, the transient solution of this problem is described as the linear combination of the general solutions. Consequently, we may yield the following series expansion. \[ \begin{array}{ll}h(x,t)=&A_{01}e^{\sqrt{\frac{L}{T}x}}+B_{01}e^{-\sqrt{ \frac{L}{T}x}}+A_{02}xe^{-\frac{L}{S}t}+B_{02}e^{-\frac{L}{S}t}\\ &+\sum\limits_{p=1}^{w}\left[A_{1p}(\frac{T^{p2}-L}{S})t+px+B_{1p}e^{(\frac{T ^{p2}-L}{S})t-px}\right]\\ &+\sum\limits_{p=1}^{w}\left[A_{2p}e^{-(\frac{T^{p2}+L}{S})t}\cos(px)+B_{2p}e ^{-(\frac{T^{p2}+L}{S})t}\sin(px)\right]\end{array} , \tag{9}\]  
where \(w\) denotes the basis function order, \(A_{01}\), \(B_{01}\), \(A_{02}\), \(B_{02}\), \(A_{1p}\), \(B_{1p}\), \(A_{2p}\) and \(B_{2p}\) are unknowns to be found. From Equation (9), the complete basis functions, \(\mathbf{T}\), are acquired as  
\[\mathbf{T}=\bigg{\{}e^{\sqrt{\frac{L}{T}x}},e^{-\sqrt{\frac{L}{T}x}},xe^{- \frac{L}{S}t},e^{-\frac{L}{S}t},e^{(\frac{T^{p2}-L}{S})t+px},e^{(\frac{T^{p2} -L}{S})t-px},e^{-(\frac{T^{p2}+L}{S})t}\cos(px),e^{-(\frac{T^{p2}+L}{S})t}\sin (px)\bigg{\}}. \tag{10}\]The Neumann boundary is given as  
\[ \begin{array}{ll}\frac{\partial t}{\partial x}=&A_{01}\sqrt{\frac{t}{x}}e^{ \sqrt{\frac{t}{x}}x}+B_{01}\Big{(}-\sqrt{\frac{t}{t}}\Big{)}e^{-\sqrt{\frac{t}{ x}}x}+A_{02}e^{-\frac{t}{5}t}+B_{02}\\ &+\sum\limits_{p=1}^{w}\left[A_{1p}pe^{(\frac{T^{2}-t}{s})t+px}+B_{1p}(-p)e^{( \frac{T^{2}-t}{s})t-px}\right]\\ &+\sum\limits_{p=1}^{w}\left[-A_{2p}pe^{-(\frac{T^{2}+t}{s})t}\sin(px)+B_{2p} pe^{-(\frac{T^{2}+t}{s})t}\cos(px)\right]\end{array} . \tag{11}\]  
To evaluate the coefficients in Equations (9) and (11), the collocation method is utilized. We may assign collocation points on the Dirichlet and Neumann boundary using Equations (9) and (11). Then, the system of linear algebraic equations may be assembled as follows. \[ \begin{array}{llll}\left[\begin{array}{cccc}e^{\sqrt{\frac{t}{x}}x_{1}}&e^ {-\sqrt{\frac{t}{x}}x_{1}}&e^{(\frac{T^{2}-t}{2})t+px_{1}}&\ldots&e^{-(\frac{T ^{2}+t}{2})t+px}\sin(px_{1})\\ e^{\sqrt{\frac{t}{x}}x_{2}}&e^{-\sqrt{\frac{t}{x}}x_{2}}&e^{(\frac{T^{2}-t}{2})t+ px_{2}}&\ldots&e^{-(\frac{T^{2}+t}{2})t+px}\sin(px_{2})\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ e^{\sqrt{\frac{t}{x}}x_{1+x_{2}}}&e^{(\frac{T^{2}+t}{2})t+px}&e^{(\frac{T^{2} +t}{2})t+px_{1}+x_{2}}&\ldots&e^{-(\frac{T^{2}+t}{2})t+px}\sin(px_{1+x_{2}})\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ \sqrt{\frac{t}{x}}e^{(\frac{t}{x}_{2x_{1+x}}a_{Rx}}&-\sqrt{\frac{t}{x}}e^{- \sqrt{\frac{t}{x}}x_{3x_{1+x}}a_{Rx}}&pe^{(\frac{T^{2}-t}{2})t+px_{1}+x_{2}}a _{Rx}&\ldots&pe^{-(\frac{T^{2}+t}{2})t+px_{1}+x_{2}}\cos(px_{2x_{1+x}})n_{x} \end{array}\right]\end{array}