## 2 Compact Difference Schemes
### Truncation Error  
As previously mentioned, the CD scheme is said to be of the \(p\)th order of accuracy if its truncation error is proportional to \(h^{p}\). According to Formula (3), for a given \(M\) and \(N\), the approximations of the derivatives of different orders have a different order of accuracy. This is a direct result of the expansion of the function \(f\) and its derivatives \(f^{(r)}\) in (1) into the Taylor series. For example, for \(M=1\) and \(N=3\), the Taylor series expansions of the left-hand side of the Formula (1) are the following:  
* The 1 st derivative approximation: \[f_{i}^{\prime}+2\,a_{1}\bigg{(}f_{i}^{\prime}+\frac{h^{2}}{2!}f_{i}^{(3)}+\frac{ h^{4}}{4!}f_{i}^{(5)}+\frac{h^{6}}{6!}f_{i}^{(7)}+\mathcal{O}\Big{(}h^{8} \Big{)}\bigg{)}\] (8)
* The 3 rd derivative approximation: \[f_{i}^{(3)}+2\,a_{1}\bigg{(}f_{i}^{(3)}+\frac{h^{2}}{2!}f_{i}^{(5)}+\frac{h^{4} }{4!}f_{i}^{(7)}+\mathcal{O}\Big{(}h^{6}\Big{)}\bigg{)}\] (9)
* The 5 th derivative approximation: \[f_{i}^{(5)}+2\,a_{1}\bigg{(}f_{i}^{(5)}+\frac{h^{2}}{2!}f_{i}^{(7)}+\mathcal{O }\Big{(}h^{4}\Big{)}\bigg{)}\] (10)  
This shows that using (3) for \(M=1\) and \(N=3\), we obtain the 8 th order scheme for the 1 st derivative approximation, the 6 th order scheme for the 3 rd derivative, and the 4 th order scheme for the 5 th derivative. Hence, to have the same order of accuracy for all derivatives, one has to use different values of \(M\) and \(N\) for particular derivatives. For example, the 6 th order of accuracy is achieved with \(M=1\) and \(N=2\) for the 1 st derivative, \(M=1\) and \(N=3\) for the 3 rd derivative, \(M=1\) and \(N=4\) for the 5 th derivative. In this paper, we apply the first approach with the same \(M\) and \(N\) for all derivatives occurring in particular PDEs. This is motivated by the fact that only in this case is the computational cost of the solution of (2) the same for derivatives of different orders. It is worth noting that CD schemes with different \(M\) and \(N\) values may have the same order of accuracy. For instance, \(M=1\), \(N=7\) and \(M=2\), \(N=6\) and \(M=3\), \(N=5\) as well as \(M=4\), \(N=4\) result in the 16 th-order scheme for the 1 st and 2 nd derivative approximation. One should mention that changing \(M\) and \(N\) while keeping the same order of approximation has very important consequences, i.e., the leading order of the truncation error changes. For instance, the truncation error of the 16 th-order scheme of the 1 st derivative approximation with \(M=1\) and \(M=7\) is equal to \(\frac{-25,401,600}{17!}h^{16}f_{i}^{(17)}\), whereas with \(M=2\) and \(M=6\), it drops to \(\frac{-2,073,600}{17!}h^{16}f_{i}^{(17)}\), then with \(M=3\) and \(N=5\), it is equal to \(\frac{-518,400}{17!}h^{16}f_{i}^{(17)}\), and with \(M=4\) and \(N=4\), it is \(\frac{-331,726}{17!}h^{16}f_{i}^{(17)}\). At the same time, applying the same values of \(M\) and \(N\) for the approximation of 3 rd and 4 th-order derivatives leads to the 14 th-order scheme, and the approximation of the 5 th and 6 th-order derivatives is of the 12 th order. Hence, for a given \(M\) and \(N\), the expected order of the solution accuracy of a given PDE is determined by the order of approximation of the highest-order derivative occurring in this equation.