## 3 The Proposed Meshless Numerical Scheme
### Space-Time Generalized Finite Difference Method  
In this subsection, the numerical process of ST-GFDM is described. Once the time-dependent problem is determined, the distribution nodes can be set in the corresponding computational domain. For the one-dimensional (1D) time-dependent problem, the ST approach defines a two-dimensional (2D) steady-state problem in an \(x\)-\(t\) field. Since the ST approach is applied, the distribution nodes are set both in the space- and time-axes and named ST-domain [38; 40; 41], as shown in Figure 1a. A supporting domain is set up by choosing the \(n_{\mathrm{s}}\) nearest nodes within the central \(i\) node in Figure 1b (see the \"\(\times\)\" symbol), and \((x_{i},t_{i}),\ i=1,2,\ldots,N_{T}\) is defined as the coordinate of the \(i\)th node. On the other hand, \(\big{(}x_{i,j},t_{i,j}\big{)},\ j=1,2,\ldots,n_{\mathrm{s}}\) represent the coordinates of the nodes in the supporting domain. By applying the \(n\)-order Taylor expansion and the weighting function, a residual function \(B_{n}(u_{i})\) can be written as [36; 51],  
\[B_{n}(u_{i})=\sum_{j=1}^{n_{\mathrm{s}}}\left[\left(-u_{i,j}+\sum_{0\leq o,m \leq n,\ o+m\leq n}\frac{1}{o!m!}\cdot\frac{\partial^{o+m}u_{i}}{\partial^{o} x\partial^{m}t}\cdot h_{i,j}^{o}\cdot l_{i,j}^{m}+E_{n}\right)w\big{(}h_{i,j},l_{i,j} \big{)}\right]^{2}, \tag{9}\]  
where \(u_{i}\) and \(u_{i,j}\) denotes the unknowns at the \(i\)th node and unknowns inside the \(i\)th supporting domain, respectively; \(E_{n}\) is the truncation error of the \(n\)-order Taylor expansion; o and m is the natural number; \(h_{i,j}=x_{i}-x_{i,j}\); \(l_{i,j}=t_{i}-t_{i,j}\); and \(w\) is the quartic spline function and describes as follows:  
\[w\big{(}h_{i,j},l_{i,j}\big{)}= \begin{cases}1-6\Big{(}\frac{d_{i,j}}{dm_{i}} \Big{)}^{2}+8\Big{(}\frac{d_{i,j}}{dm_{i}}\Big{)}^{3}-3\Big{(}\frac{d_{i,j}}{ dm_{i}}\Big{)}^{4},d_{i,j}\leq dm_{i}\\ 0\hskip 142.26378 pt,d_{i,j}>dm_{i}\end{cases} , \tag{10}\]  
where \(d_{i,j}\) is the distance between the \(i\)th node and each node inside the \(i\)th supporting domain, and \(d_{i,j}=\sqrt{h_{i,j}^{2}+l_{i,j}^{2}}\); and \(dm_{i}\) is the maximum distance. In this study, the 3 rd-order ST-GFDM was applied, and Equation (9) can be written as,  
Figure 1: The schematic diagram of (**a**) nodes and (**b**) a supporting domain of the GFDM in the \(x\)-\(t\) domain. \[B_{3}(u_{i})=\sum\limits_{j=1}^{n_{s}}\left[\left( \begin{array}{c}u_{i}-u_{i,j}+ h_{ij}\frac{\partial u}{\partial x}\Big{|}_{i}+l_{ij}\frac{\partial u}{ \partial x}\Big{|}_{i}+\frac{h_{ij}^{2}}{2}\frac{\partial^{2}u}{\partial x^{2} }\Big{|}_{i}+h_{ij}l_{ij}\frac{\partial^{2}u}{\partial x\partial t}\Big{|}_{i}+ \frac{l_{ij}^{2}}{2}\frac{\partial^{2}u}{\partial x^{2}}\Big{|}_{i}+\cdots\\ \frac{h_{ij}^{2}}{6}\frac{\partial^{2}u}{\partial x^{3}}\Big{|}_{i}+\frac{h_ {ij}^{2}l_{ij}}{2}\frac{\partial^{3}u}{\partial x^{2}ot}\Big{|}_{i}+\frac{h_ {ij}l_{ij}}{2}\frac{\partial^{3}u}{\partial x\partial t}\Big{|}_{i}+\frac{h_ {ij}}{6}\frac{\partial^{3}u}{\partial x^{3}}\Big{|}_{i}+E_{3}\end{array} \right) w_{i,j}\right]^{2}, \tag{11}\]  
The order of approximations in Equation (11), which accounts for the truncation errors, is determined by the Taylor series expansion up to the 3 rd order.