# Construction of Compact Finite Difference Schemes by Classic Differential Quadrature
## 3 Construction of Compact Difference Schemes by DQM
### The Extended Trapezoidal Rules  
#### 3.4.1 Extended Trapezoidal Rules of the First Kind  
As is shown in Figure 1, and let \(s\) be an odd number. Define  
\[v=(s+1)/2 \tag{31}\]  
then we can obtain the following formula via DQM  
\[\mathbf{G}\left[ \begin{array}{c}y_{n}\\ \vdots\\ y_{n+v-1}\\ y_{n+v}\\ \vdots\\ y_{n+s}\end{array} \right]=h\left[ \begin{array}{c}f_{n}\\ \vdots\\ f_{n+v-1}\\ f_{n+v}\\ \vdots\\ f_{n+s}\end{array} \right],\ \mathbf{G}\in R^{(s+1)\times(s+1)} \tag{32}\]  
The same as Equation (12), Equation (32) is also the basic calculation scheme of DQM, however, the scheme of Equation (12) is more widely used in engineering application. In general, the matrix \(\mathbf{G}\) in Equation (32) is not a full rank matrix, so its inverse matrix does not exist. If the distribution of grid points is centro-symmetrical in the normalized interval \([0,\,1]\), the matrix \(\mathbf{G}\) in Equation (32) will be a centro-antisymmetrical matrix. Since the inverse matrix of \(\mathbf{G}\) does not exist, it is natural to consider its generalized inverse matrix. As is well known, the generalized inverse matrix--namely, the pseudo-inverse matrix \(\mathbf{G}^{+}=\left(\mathbf{G}^{\intercal}\mathbf{G}\right)^{-1}\mathbf{G}^{\intercal}\) of centro-antisymmetrical matrix with even order--will also be a centro-antisymmetrical matrix. Define  
\[\mathbf{G}^{+}=\mathbf{A}\equiv\left[a_{pq}\right],\ p,q\in(1,\ s+1) \tag{33}\]  
then the Equation (32) can be written as  
\[\left[ \begin{array}{c}y_{n}\\ \vdots\\ y_{n+v-1}\\ y_{n+v}\\ \vdots\\ y_{n+s}\end{array} \right]=h\mathbf{A}\left[ \begin{array}{c}f_{n}\\ \vdots\\ f_{n+v-1}\\ f_{n+v}\\ \vdots\\ f_{n+s}\end{array} \right] \tag{34}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(s\) & \(v\) & \(\beta_{0}\) & \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) & \(\beta_{4}\) & \(\beta_{5}\) & \(\beta_{6}\) \\ \hline
2 & 1 & \(\frac{5}{360}\) & \(\frac{2}{6}\) & \(-\frac{1}{36}\) & & & & & \\
4 & 2 & \(-\frac{370}{360}\) & \(\frac{173}{1800}\) & \(\frac{173}{1800}\) & \(-\frac{37}{1800}\) & \(\frac{11}{3600}\) & & & \\
6 & 3 & \(\frac{271}{423}\) & \(\frac{1023}{3528}\) & \(\frac{10273}{141,1230}\) & \(\frac{586}{6615}\) & \(-\frac{37}{2257}\) & \(\frac{67}{77,840}\) & \(-\frac{191}{423,360}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Coefficients of generalized Adams methods (GAMs) using uniform grids (for even \(s\)). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(s\) & \(v\) & \(\beta_{0}\) & \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) & \(\beta_{4}\) & \(\beta_{5}\) & \(\beta_{6}\) & \(\beta_{7}\) \\ \hline
1 & 1 & \(\frac{1}{4}\) & \(\frac{1}{4}\) & & & & & & \\
3 & 2 & \(-\frac{1}{96}\) & \(\frac{13}{96}\) & \(\frac{13}{96}\) & \(-\frac{1}{96}\) & & & & & \\
5 & 3 & \(\frac{11}{8450}\) & \(-\frac{31}{8280}\) & \(\frac{401}{423}\) & \(\frac{401}{423}\) & \(-\frac{31}{8280}\) & \(\frac{11}{8643}\) & & & \\
7 & 4 & \(-\frac{191}{967,800}\) & \(\frac{1857}{967,800}\) & \(-\frac{355}{35,840}\) & \(\frac{68,523}{767,600}\) & \(\frac{632}{967,680}\) & \(-\frac{191}{35,840}\) & \(\frac{1879}{967,680}\) & \(-\frac{191}{967,6800}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Coefficients of GAMs using uniform grids (for odd \(s\)). Subtracting the \(v\)-th equation from the \((v+1)\)-th equation in the matrix Equation (34), we can obtain the following equation  
\[y_{n+v}-y_{n+v-1}=h\sum_{z=0}^{s}\rho_{z}f_{n+z} \tag{35}\]  
It is obvious that the aforementioned Equation (35) is the GAMs with \((s+1)\)-order, in which  
\[\rho_{z}=a_{v+1,z+1}-a_{v,z+1},\;z\in(0,\;s) \tag{36}\]  
Considering that \(s=2v-1\) and \(A\) defined in Equation (33) is a centro-antisymmetrical matrix with even dimension, it can be easily proved that \(\rho_{z}\) in Equation (36) is centro-symmetrical.