# Construction of Compact Finite Difference Schemes by Classic Differential Quadrature
## 3 Construction of Compact Difference Schemes by DQM
### The Generalized Adams Methods  
As is shown in Figure 2, let \(x_{n-1}<\widetilde{x}<x_{n}\), and divide the interval \(x\in[\widetilde{x},x_{n+1}]\) into \(s+1\) segments, in which \(h=x_{n+1}-\widetilde{x}\), \(x(\widetilde{x}+c_{0}h)\equiv\widetilde{x}\), \(x(\widetilde{x}+c_{s}h)\equiv x_{1}\), \(x(\widetilde{x}+c_{s+1}h)\equiv x_{n+1}\).  
Define  
\[v=\left\{\begin{array}{l}s/2,\;\text{for even}\;s\\ (s+1)/2,\;\text{for odd}\;s\end{array}\right. \tag{22}\]  
then we can obtain the following formula according to Equation (12)  
\[\widetilde{G}\left[\begin{array}{c}y_{n}\\ \vdots\\ y_{n+v-1}\\ y_{n+v}\\ \vdots\\ y_{n+s}\end{array}\right]+G_{0}y(\widetilde{x})=h\left[\begin{array}{c}f_{n }\\ \vdots\\ f_{n+v-1}\\ f_{n+v}\\ \vdots\\ f_{n+s}\end{array}\right] \tag{23}\]  
in which \(\widetilde{G}\in R^{(s+1)\times(s+1)}\), \(G_{0}\in R^{(s+1)\times 1}\); \(y_{n+z}=y(\widetilde{x}+c_{z+1}h)\), \(z\in(0,\;s)\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(s\) & \(v\) & \(\alpha_{0}\) & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(\alpha_{4}\) & \(\alpha_{5}\) & \(\alpha_{6}\) & \(\alpha_{7}\) & \(\alpha_{8}\) \\ \hline
2 & 1 & \(-1\) & 0 & 1 & & & & & & \\
4 & 2 & \(\frac{1}{3}\) & \(-\frac{8}{3}\) & 0 & \(\frac{8}{3}\) & \(-\frac{1}{3}\) & & & & & \\
6 & 3 & \(-\frac{1}{10}\) & \(\frac{10}{10}\) & \(-\frac{9}{2}\) & 0 & \(\frac{9}{2}\) & \(-\frac{9}{10}\) & \(\frac{1}{10}\) & & & \\
8 & 4 & \(\frac{1}{35}\) & \(-\frac{32}{105}\) & \(\frac{8}{5}\) & \(-\frac{32}{5}\) & 0 & \(\frac{32}{5}\) & \(-\frac{8}{5}\) & \(\frac{32}{105}\) & \(-\frac{1}{35}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The coefficients of central difference schemes using uniform grids.  
In additional, define  
\[\widetilde{\mathbf{A}}=\widetilde{\mathbf{G}}^{-1}\equiv\left[\widetilde{a}_{pq}\right],\ p,q\in(1,\,s+1) \tag{24}\]  
Since it has been proved that [12; 15]  
\[\mathbf{G}_{0}=-\widetilde{\mathbf{G}}\mathbf{e} \tag{25}\]  
where, \(\mathbf{e}\) is the unit column vector, therefore, the Equation (23) can be rewritten as  
\[\left[\begin{array}{c}y_{n}\\ \vdots\\ y_{n+v-1}\\ y_{n+v}\\ \vdots\\ y_{n+s}\end{array}\right]=y(\widetilde{x})\mathbf{e}+h\widetilde{\mathbf{A}}\left[ \begin{array}{c}f_{n}\\ \vdots\\ f_{n+v-1}\\ f_{n+v}\\ \vdots\\ f_{n+s}\end{array}\right] \tag{26}\]  
Subtracting the \(v\)-th equation from the \((v+1)\)-th equation in the matrix Equation (26) yields the following equation  
\[y_{n+v}-y_{n+v-1}=h\sum_{z=0}^{s}\beta_{z}f_{n+z} \tag{27}\]  
where,  
\[\beta_{z}=\widetilde{a}_{v+1,z+1}-\widetilde{a}_{v,z+1},\ z\in(0,\,s) \tag{28}\]  
It can be shown that the above numerical method (27) has the accuracy of \((s+1)\)-order.  
The numerical method (27) has a similar form to the traditional Adams methods, but we will later show that this method has the better numerical stability then traditional Adams methods. We call this numerical method the generalized Adams methods (GAMs).  
When selecting the uniform grids, the coefficient values (\(\beta_{z}\)) of GAMs are shown in the following Tables 4 and 5 respectively.  
For the uniform grids, we can define that  
\[\overline{h}=\left(x_{n+1}-\widetilde{x}\right)/(s+1)=h/(s+1) \tag{29}\]  
therefore, the method (27) could be rewritten as  
\[y_{n+v}-y_{n+v-1}=\overline{h}\sum_{z=0}^{s}\ (\frac{\beta_{z}}{s+1})f_{n+z} \tag{30}\]  
According to Tables 4 and 5, we can verify that the proposed method (30) is consistent with the GAMs proposed by [PERSON] et al. [19]. In reference [21], it has been shown that, for even \(s\), the methods (30) are both A-stable, and for odd \(s\) the methods (30) are also A-stable while the stability region is just the complete left half plane.  
Figure 2: Schematic diagram of the grid points for generalized Adams methods (GAMs).