## 2 Methodology
### GFDM for Spatial Partial Derivative Approximation  
In the GFDM, through utilizing the Taylor series expansion and weighted least squares fitting, the derivatives of unknown variables can be approximated by a linear combination of function values of some neighboring nodes, which are located inside a star. First, a regular or irregular cloud of points is generated in the computation domain and along the boundary. For a given node \(i\), which is denoted as a central node, the \(m\) nearest nodes surrounding the node \(i\) will be found. The concept of the star refers to a group of established nodes in relation to the central node, as shown in the black circle in Figure 1. Each node is assigned an associated star. If \(u_{0}\) is the function value at the central node \(x_{0}\) of the star and \(u_{i}(i=1,\ 2,\ \cdots,\ m)\) is the function value at one of the rest of the nodes, then the Taylor series expansion around the central node can be expressed in the following form:  
\[u_{i}=u_{0}+h_{i}\frac{\partial u_{0}}{\partial x}+l_{i}\frac{\partial u_{0}}{ \partial z}+\frac{h_{i}^{2}}{2}\frac{\partial^{2}u_{0}}{\partial x^{2}}+\frac{ l_{i}^{2}}{2}\frac{\partial^{2}u_{0}}{\partial z^{2}}+h_{i}l_{i}\frac{\partial^{2}u_{0}}{ \partial x\partial z}+\cdots \tag{7}\]  
where the coefficients \(h_{i}=x_{i}-x_{0}\), \(l_{i}=z_{i}-z_{0}\), and \((x_{0},z_{0})\) are the coordinates of the central node, and \((x_{i},z_{i})\) are the coordinates of the node \(i\) in the star. By truncating the Taylor series with the second derivative, the residual function \(B(u)\) can be defined by the following equation:  
\[B(u)_{2^{nd}}=\sum\limits_{i=1}^{m}\left[\left(u_{0}-u_{i}+h_{i}\frac{ \partial u_{0}}{\partial x}+l_{i}\frac{\partial u_{0}}{\partial z}+\frac{h_{ i}^{2}}{2}\frac{\partial^{2}u_{0}}{\partial x^{2}}+\frac{l_{i}^{2}}{2}\frac{ \partial^{2}u_{0}}{\partial z^{2}}+h_{i}l_{i}\frac{\partial^{2}u_{0}}{ \partial x\partial z}\right)w_{i}\right]^{2} \tag{8}\]  
By truncating the Taylor series with the fourth derivative, the residual function \(B(u)\) can be defined by the following:  
\[ \begin{array}{l}B(u)_{4^{th}}=\sum\limits_{i=1}^{m}\left[\left(u_{0}-u_{i}+ h_{i}\frac{\partial u_{0}}{\partial x}+l_{i}\frac{\partial u_{0}}{\partial z }+\frac{h_{i}^{2}}{2}\frac{\partial^{2}u_{0}}{\partial z^{2}}+\frac{l_{i}^{2} }{2}\frac{\partial^{2}u_{0}}{\partial z^{2}}+h_{i}l_{i}\frac{\partial^{2}u_{0 }}{\partial x\partial z}\right.\\ \left.+\frac{h_{i}^{2}}{6}\frac{\partial^{2}u_{0}}{\partial x^{2}}+\frac{l_{i} ^{2}}{6}\frac{\partial^{2}u_{0}}{\partial z^{2}}+\frac{h_{i}^{2}l_{i}}{2} \frac{\partial^{2}u_{0}}{\partial x^{2}}+\frac{h_{i}^{2}l_{i}^{2}}{2}\frac{ \partial^{2}u_{0}}{\partial x\partial z}+\frac{h_{i}^{2}}{24}\frac{\partial^{ 2}u_{0}}{\partial x^{2}}+\frac{l_{i}^{2}}{24}\frac{\partial^{2}u_{0}}{\partial x ^{4}}+\frac{l_{i}^{2}}{24}\frac{\partial^{4}u_{0}}{\partial z^{4}}\\ \left.+\frac{h_{i}^{2}l_{i}}{6}\frac{\partial^{2}u_{0}}{\partial x^{2}\partial z }+\frac{h_{i}^{2}l_{i}^{2}}{6}\frac{\partial^{2}u_{0}}{\partial x\partial z^{ 2}}+\frac{h_{i}^{2}l_{i}^{2}}{4}\frac{\partial^{2}u_{0}}{\partial x^{2} \partial z^{2}}w_{i}\right]^{2}\right]\end{array}  \tag{9}\]  
In Equations (8) and (9), \(w_{i}=w(h_{i},l_{i})\) represents the distance weight function of the star. In all the examples considered in this paper, the weighting function was chosen as per [27]:  
\[w_{i}= \begin{cases}1-6\left(\frac{d_{i}}{d_{\max}}\right)^{2}+8\left(\frac{d_{ i}}{d_{\max}}\right)^{3}-3\left(\frac{d_{i}}{d_{\max}}\right)^{4},&d_{i}\leq d_{\max}\\ 0,&d_{i}>d_{\max}\end{cases}  \tag{10}\]  
where \(d_{i}\) denotes the distance between nodes \((x_{i},z_{i})\) and \((x_{0},z_{0})\), and \(d_{\max}\) is the distance between the central node and the farthest node in the star.