## 2 Compact Difference Schemes
### Fourier Analysis  
In this section, we present a Fourier analysis of the differentiation errors. It shows how the scales of a given wavelength are represented by a particular discretization scheme. A periodic function \(f\) in the domain \([0,L]\) can be expressed by the Fourier series as:  
\[f(x)=\sum_{n=-K/2}^{K/2}\hat{f}_{n}\exp\bigg{(}i\frac{2\pi nx}{L}\bigg{)}= \sum_{n=-K/2}^{K/2}\hat{f}_{n}\exp(i\omega_{n}s) \tag{11}\]  
where \(i=\sqrt{-1}\), \(\omega_{n}=2\pi nh/L=2\pi n/K\) is the scaled wave number in the range \([0,\pi]\) and \(s=x/h\) is the scaled coordinate. The \(r\)-th derivative of an arbitrary Fourier mode (\(\hat{f}\exp(i\omega s)\)) with respect to \(s\) is equal to \(f^{(r)}=(i\omega)^{r}\hat{f}\exp(i\omega s)\), while its approximate counterpart is \(f_{\text{\tiny{RD}}}^{(r)}=(i\omega^{r})^{r}\hat{f}\exp(i\omega s)\), where \(\omega^{r}\) is a modified wave number. At the nodal points \((\ldots,j-1,j,j+1\ldots)\), the function values and the derivatives are equal to \(f^{(r)}_{\text{n}j-1}=(i\omega^{\prime})^{r}\hat{f}\exp(i\omega(s-1)),f^{(r)}_{ \text{n}j}=(i\omega^{\prime})^{r}\hat{f}\exp(i\omega s))\), \(f^{(r)}_{\text{n}j+1}=(i\omega^{\prime})^{r}\hat{f}\exp(i\omega(s+1))\), etc. Introducing these expressions into the approximation formulas leads to the following relationships:  
\[\omega^{(r)}(\omega)=\hat{t}^{r-1}\cdot\frac{2\sum\limits_{j=1}^{N}\sin(j \omega)b_{j}}{1+2\sum\limits_{k=1}^{M}a_{k}\cos(k\omega)},\qquad\omega^{(r)}( \omega)=\hat{t}^{r}\cdot\frac{2\sum\limits_{j=1}^{N}(\cos(j\omega)-1)b_{j}}{1+ 2\sum\limits_{k=1}^{M}a_{k}\cos(k\omega)} \tag{12}\]  
for odd-order and even-order derivatives, respectively. As an example, Figures 1 and 2 show the modified wave numbers of the 3 rd and 4 th derivative approximations for various \(M\) and \(N\). The exact Fourier differentiation corresponds to the solid lines. As one can see, for large \(M\) or \(N\), the values of \(\omega^{\prime\prime\prime}\) and \(\omega^{\prime\prime\prime\prime}\) are close to \(\omega\) over a wide range. They only differ when \(\omega\) is large, which means that small scale phenomena corresponding to this range of \(\omega\) are poorly resolved. A few CD schemes seem to be peculiar, for example the schemes with \(M=1\) and \(M=2\), \(M=2\) and \(N=5\), as well as with \(M=4\) and \(N=7\) for the 3 rd derivative approximation and schemes with \(M=2\) and \(N=2\) for the 4 th derivative pronouncedly overshoot the exact values of the wave numbers. Very similar behavior was reported in [3]. Surprisingly, these schemes do not lead to an erroneous solution. One can also see that in the case of third-order derivative approximation, the 14 th-order scheme with \(M=2\) and \(N=6\) has a better resolving efficiency of small scales than 16 th-order scheme with \(M=2\) and \(N=7\). Similar behavior can be observed for the approximation of the fourth-order derivative with CD schemes with \(M=4\) and \(N=7\) and \(N=8\). In this case, the 20 th-order scheme was revealed to be more accurate than the 22 nd-order scheme.  
Figure 1: Modified wave number of the 3 rd derivative approximation for various CD schemes. (**a**) \(M=1\) and different \(N\), (**b**) \(M=2\) and different \(N\), (**c**) \(M=3\) and different \(N\), (**d**) \(M=4\) and different \(N\).