## 2 Compact Difference Schemes  
The problems analyzed in this paper are periodic and one-dimensional. Methods of deriving boundary schemes for non-periodic problems that could be applied to extend the scope of the present analysis are based on tuned explicit one-sided formulas [51] or summation-by-parts and simultaneous-approximation-term methods analyzed in detail in [52; 53]. We assume a uniform computational mesh on the periodic interval \([0,L]\), consisting of \(K\) nodes \(x_{i}=h(i-1)\) for \(1\leq i\leq K\), with the mesh size \(h=L/K\). After some algebra, general formulas for CD schemes for the approximation of the odd-order (\(r=1,3,5,\ldots\)) and even-order (\(r=2,4,6,\ldots\)) derivatives \(\partial^{r}f/\partial x^{r}\) of the function \(f\) can be defined as:  
\[ \begin{cases}f_{i}^{(r)}+\sum\limits_{k=1}^{M}a_{k}\Big{(}f_{i-k}^{(r)}+f_{i+ k}^{(r)}\Big{)}&=\frac{1}{H^{r}}\sum\limits_{j=1}^{N}b_{j}\big{(}f_{i+j}-f_{i-j} \big{)},&\text{for}\;\;r=1,3,5,\ldots\\ f_{i}^{(r)}+\sum\limits_{k=1}^{M}a_{k}\Big{(}f_{i-k}^{(r)}+f_{i+k}^{(r)}\Big{)}&= \frac{1}{H^{r}}\sum\limits_{j=1}^{N}b_{j}\big{(}f_{i+j}-2f_{i}+f_{i-j}\big{)}, &\text{for}\;\;r=2,4,6,\ldots\end{cases}  \tag{1}\]  
The number of coefficients \(a_{k}\) and \(b_{j}\) depend on the number of points \(M\) and \(N\), with \(k=1,\ldots,M,j=1,\ldots,N\). These coefficients are calculated on the basis of expanding the function \(f\) and its derivative \(f^{(r)}\) into the Taylor series and equating the appropriate terms on the left and right sides of the resulting equations. Writing Equation (1) for all mesh points leads to a linear system of equations of which the solution gives the vector of derivative \(\mathbf{f}^{(\mathbf{r})}\) simultaneously in all nodes. This system and its solution have the following form:  
\[\mathbf{A}\mathbf{f}^{(\mathbf{r})}=\frac{1}{h^{r}}\mathbf{B}\mathbf{f}\quad \rightarrow\quad\mathbf{f}^{(\mathbf{r})}=\frac{1}{h^{r}}\underbrace{\mathbf{A }^{-1}_{\mathbf{D}}\mathbf{B}}_{\mathbf{D}}\mathbf{f} \tag{2}\]  
where \(\mathbf{A}\) and \(\mathbf{B}\) are matrices of the dimension \(K\) composed of the coefficients \(a_{k}\) and \(b_{j}\), respectively. For the FD method, the coefficients \(a_{k}\) are equal to zero and the matrix \(\mathbf{A}\) is the identity matrix. In this paper, we consider the approximation formulas with large \(M\) and \(N\geq M\), which can be at most half the points on the mesh, i.e., \(K/2\). In this case, the matrices \(\mathbf{A}\) and \(\mathbf{B}\) are dense and fully filled by \(a_{k}\) and \(b_{j}\). However, one should note that if \(M\) is small, the matrix \(\mathbf{A}\) in (2) is sparse, e.g., for \(M=1\), \(2\), \(3\) it would be tri-, five-, or seven-diagonal. In these cases, instead of solving the system (2) by computing the inverse matrix \(\mathbf{A}^{-1}\) an efficient algorithm for solving sparse systems of linear equations can be employed. The order of the approximation \(p\) in \(\mathcal{O}(h^{p})\) is equal to  
\[p=\left[(2M+2N)-(r-s)\right] \tag{3}\]  
for odd-order (\(s=1\)) and for even-order (\(s=2\)) derivatives. Here, we present a general formula for determining the coefficients \(a_{k}\) and \(b_{j}\) for an arbitrary CD scheme from the systems of \(M+N\) equations:  
\[ \begin{cases}1+2\sum\limits_{k=1}^{M}a_{k}=\frac{2}{r}\sum\limits_{j=1}^{N}j^ {r}b_{j},\\ \sum\limits_{k=1}^{M}\left(\prod\limits_{l=1}^{r}(2i+l)\right)k^{2i}a_{k}= \sum\limits_{j=1}^{N}j^{2i+r}b_{j},&\text{for }\;i=1,\ldots,N+M-\frac{r+1}{2},\\ \sum\limits_{j=1}^{N}j^{2i-1}b_{j}=0,&\text{for }\;i=1,\ldots,\frac{r+1}{2}-1. \end{cases}  \tag{4}\]  
for odd-order derivatives (\(r=1,3,5,\ldots\)) and:  
\[ \begin{cases}1+2\sum\limits_{k=1}^{M}a_{k}=\frac{2}{r}\sum\limits_{j=1}^{N}j^ {r}b_{j},\\ \sum\limits_{k=1}^{M}\left(\prod\limits_{l=1}^{r}(2i+l)\right)k^{2i}a_{k}= \sum\limits_{j=1}^{N}j^{2i+r}b_{j},&\text{for }\;i=1,\ldots,N+M-\frac{r}{2},\\ \sum\limits_{j=1}^{N}j^{2i}b_{j}=0,&\text{for }\;i=1,\ldots,\frac{r}{2}-1. \end{cases}  \tag{5}\]  
for even-order derivatives (\(r=2,4,6,\ldots\)).