## 3 Methodology
### RBF-FD Approximations
\begin{table}
\begin{tabular}{l l} \hline \hline
**Type of RBFs** & **Definition \(\boldsymbol{\phi}\)(r)** \\ \hline Infinitely smooth RBFs & \\ Gaussian (GA) & \(\phi(r,\varepsilon)=e^{-(cr)^{2}}\) \\ Inverse multiquadric (IMQ) & \(\phi(r,\varepsilon)=\frac{1}{\sqrt{1+(cr)^{2}}}\) \\ Inverse quadratic (IQ) & \(\phi(r,\varepsilon)=\frac{1}{1+(cr)^{2}}\) \\ Multiquadric (MQ) & \(\phi(r,\varepsilon)=\sqrt{1+(cr)^{2}}\) \\ \hline Piecewise smooth RBFs & \\ Polyharmonic spline (PHS) & \(\phi(r)=\left\{\begin{array}{ll}r^{2n+1}\\ r^{2n}ln(r)\;n=1,2,3,4,5,\ldots\\ \text{Compact support}\end{array}\right.\) & \(\phi(r)=(1-r)_{+}^{m}p(r)\;m=1,2,3,4,5,\ldots\) \\ \hline Other hybrid RBFs & \\ Hybrid Gaussian-cubic & \(\phi(r)=\alpha e^{-(cr)^{2}}+\beta r^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Common choices for radial basis function \(\phi\). where the RBF-FD weights \(\omega_{(1,i)}\) are formally given by the operator \(\mathcal{L}\) applied on the Lagrange form of the RBF, that is, \(\omega_{(1,i)}=\mathcal{L}_{\mathcal{X}}(\|x_{1}-x_{i}\|),\ i=1,2,\ldots n\). Moreover, it has been shown through the literature that better accuracy is gained by adding the constraint as \(\sum_{i=1}^{n}\omega_{(1,i)}=0\). Hence, the solution for the RBF-FD weights can be determined as Equation (25). \[ \begin{bmatrix}\phi(\|x_{1}-x_{1}\|)&\ldots&\phi(\|x_{1}-x_{n}\|)\\ \vdots&\ddots&\vdots\\ \phi(\|x_{n}-x_{1}\|)&\ldots&\phi(\|x_{n}-x_{n}\|)\end{bmatrix}\begin{bmatrix} \omega_{(1,1)}\\ \vdots\\ \omega_{(1,n)}\end{bmatrix}  = \begin{bmatrix}\mathcal{L}\phi(\|x_{1}-x_{1}\|)\\ \vdots\\ \mathcal{L}\phi(\|x_{1}-x_{n}\|)\end{bmatrix}  \tag{25}\]  
For the conditionally positive definite RBFs such as PHS and MQ, the polynomial terms will be included and (25) can be further represented as (26), where the last entry \(\gamma\) in the solution vector should be ignored. \[ \begin{bmatrix}&&&1&x_{1}&y_{1}\\ &\overline{A}&&\vdots&\vdots&\vdots\\ &&&1&x_{n}&y_{n}\\ 1&\cdots&1&&\\ x_{1}&\cdots&y_{1}&&1\\ y_{1}&\cdots&y_{n}&&\end{bmatrix}\begin{bmatrix}\omega_{(1,1)}\\ \vdots\\ \omega_{(1,n)}\\ -\\ \gamma_{1}\\ \gamma_{2}\\ \gamma_{3}\end{bmatrix} = \begin{bmatrix}\mathcal{L}\phi(\|x_{1}-x_{1}\|)\\ \cdots\\ \mathcal{L}\phi(\|x_{1}-x_{n}\|)\\ -\\ \mathcal{L}\lambda\|_{x-x_{1}}\\ \mathcal{L}\lambda\|_{x-x_{1}}\\ \mathcal{L}\lambda\|_{x-x_{1}}\\ \mathcal{L}\lambda\|_{x-x_{1}}\\ \mathcal{L}\lambda\|_{x-x_{1}}\end{bmatrix}  \tag{26}\]  
It can be seen that the RBF-FD weights solely depend on the relative positioning of the nodes and RBF used. Once these two parameters are defined for a particular problem, the estimation of weights can be performed in the pre-processing stage. Furthermore, as the differential operator \(\mathcal{L}\) can be arbitrary, a similar procedure can be used to obtain the weights for all function derivatives. The convention followed for denoting the weights for any point \(x_{i}\) with \(n\) supporting points is \(\omega_{i}{}^{(x)}\), \(\omega_{i}{}^{(y)}\), \(\omega_{i}{}^{(xx)}\), or \(\omega_{i}{}^{(yy)}\) when the operator \(\mathcal{L}\) is \(\frac{\partial}{\partial x}\), \(\frac{\partial}{\partial y}\), \(\frac{\partial^{2}}{\partial x^{2}}\) or \(\frac{\partial^{2}}{\partial y^{2}}\), respectively. For more details on RBF-FD discretization, refer to [23].