## 5 Numerical Aspects
### Local Generalized Differential Quadrature Method  
The Local Generalized Differential Quadrature (LGDQ) method is a numerical tool that allows to approximate the derivatives of a generic function without considering all the points of the reference domain. In other words, local interpolating basis functions are taken into account instead of their corresponding global ones, if compared to the well-known GDQ method. The main features of the LGDQ are explained in this paragraph considering a one-dimensional domain for conciseness purposes. The \(n\)-th derivative of a sufficiently smooth function \(f(x)\) evaluated at a generic point \(x_{i}\) can be computed as a weighted linear sum of the function values at some selected grid points, considering only a limited part of the whole domain, which is globally made up of \(I_{N}\) discrete points. From the mathematical point of view, one gets  
\[\left.\frac{d^{n}f(x)}{dx^{n}}\right|_{x=x_{i}}\cong\sum_{j=N_{1}}^{N_{2}}\xi_{ ij}^{(n)}f(x_{j}) \tag{55}\]  
for \(i=1,\ 2,...,\ I_{N}\), where the symbol \(\xi_{ij}^{(n)}\) denotes the weighting coefficients for derivative in hand. It is clear that the evaluation of these weighting coefficients and the choice of a proper grid distribution represent the key points of the present technique. The limit values of the sum \(N_{1},\ N_{2}\) depend on the position of the point \(x_{i}\) where the derivative is computed. The following relations must be introduced to define their value  
\[ \begin{split} N_{1}&=\max(i-I_{PN},1)\\ N_{2}&=\min(i+I_{PN},I_{N})\end{split}  \tag{56}\]  
for \(i=1,\ 2,...,\ I_{N}\). The symbol \(I_{PN}\) stands for the number of grid points that must be taken into account on the left and on the right of the point \(x_{i}\) itself. It should be noted that the value of \(I_{PN}\) has to be set a priori. Due to the definitions shown in Equation (56), the values of \(N_{1},\ N_{2}\) is variable and assumes different meanings according to the position of the point \(x_{i}\) within the domain. Figure 3 is introduced now to explain more clearly the significance of Equation (56). Let us assume that \(x_{i}\) is placed in the center of the domain so that on both its left and its right the number of points is greater than \(I_{PN}\). As it can be noted from Figure 3a, the derivative shown in Equation (55) is computed considering \((2I_{PN}+1)\) points, since \(N_{1}=i-I_{PN}\) and \(N_{2}=i+I_{PN}\). Let us consider the circumstance in which \(x_{i}\) is located next to the limits of the domain (or corresponds also to the boundary points). In these case, the value of \(I_{PN}\) can be greater than the number of points that actually appear on the left (Figure 3b) or on the right (Figure 3c) of \(x_{i}\). With reference to the case depicted in Figure 3b, one gets \(N_{1}=1\) and \(N_{2}=i+I_{PN}\), whereas \(N_{1}=i-I_{PN}\) and \(N_{2}=I_{N}\) is set for the situation shown in Figure 3c. It should be clear that the variability of \(N_{1}\), \(N_{2}\) is given not to overcome the domain boundaries. Finally, the well-known GDQ method in its global form can be obtained by setting \(N_{1}=1\) and \(N_{2}=I_{N}\) for each value of \(I_{PN}\). Figure 3: One-dimensional scheme for the Local Generalized Differential Quadrature (LGDQ) method considering different positions of the \(i\)-th point where the derivative is computed: (**a**) point placed in the center of the domain; (**b**) point located next to the limits of the domain; (**c**) point placed on the boundary. As far as the evaluation of the weighting coefficients \(\varsigma_{ij}^{(n)}\) is concerned, the formulas shown in the works [104; 123] can be used.