## 4 Conclusions  
This paper presents a detailed analysis of very high-order compact difference (CD) schemes and their accuracy when applied to solving linear and non-linear high-order partial differential equations (PDEs). The compact form of derived CD schemes for odd and even derivatives allows an easy formulation of the discretization formulas of the 20 th or 30 th order. Both the analysis of the truncation errors and the analysis in the Fourier space showed that the proposed CD schemes approach spectral accuracy in terms of the ability to resolve small-scale phenomena. The test cases included both a simple verification of the CD formulas based on the approximation of derivatives of a test function and classical problems describing important physical phenomena (wave transport, dispersion, or instability of laminar flame front), i.e., the 3 rd-order linear dispersive wave equation, 3 rd and 5 th-order Korteweg-de Vries-like equations, and 4 th-order Kuramoto-Sivashinsky equation. Comprehensive analysis showed that the obtained solutions reach the expected accuracy with the orders of the schemes in good accordance with the theoretical ones. However, it has been shown that in some cases, the increase of the computational stencil on which the CD schemes operate, and thus the increase of the approximation order, does not bring real benefits. This is because of either reaching the machine precision already on relatively coarse computational meshes or due to the round-off errors. For the CD schemes of the 16 th order or higher, both factors turned out to have a decisive impact on the reduction of the solution error, with the increase in the number of nodes in the computational meshes.  
conceptualization, L.C. and A.T.; methodology, L.C. and A.T.; software, L.C.; formal analysis, L.C. and A.T.; investigation, L.C.; data curation, L.C.; writing--original draft preparation, L.C. and A.T.; writing--review and editing, L.C. and A.T.; visualization, L.C.; funding acquisition, A.T. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Science Centre, Poland (Grant 2018/29/B/STS/00262) and statutory funds of Czestochowa University of Technology (BS/PB 1-100-3010/2022/P). The authors acknowledge [PERSON] for fruitful discussions stimulating this work that was possible thanks to the International Academic Partnerships Programme (PPI/APM/2019/1/00062) sponsored by National Agency for Academic Exchange (NAWA).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
This research was supported in part by PL-Grid Infrastructure.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time** & \(\mathbf{M=1,N=7}\) & \(\mathbf{M=2,N=6}\) & \(\mathbf{M=3,N=5}\) & \(\mathbf{M=4,N=4}\) \\ \hline
0 & \(2.70\times 10^{-13}\) & \(7.41\times 10^{-13}\) & \(3.69\times 10^{-13}\) & \(2.95\times 10^{-13}\) \\ \(\frac{\pi}{8}\) & \(6.26\times 10^{-11}\) & \(6.26\times 10^{-11}\) & \(6.26\times 10^{-11}\) & \(6.25\times 10^{-11}\) \\ \(\frac{\pi}{4}\) & \(6.26\times 10^{-11}\) & \(6.24\times 10^{-11}\) & \(6.25\times 10^{-11}\) & \(6.26\times 10^{-11}\) \\ \(\frac{\pi}{3}\) & \(2.51\times 10^{-10}\) & \(2.52\times 10^{-10}\) & \(2.52\times 10^{-10}\) & \(2.51\times 10^{-10}\) \\ \(\frac{3\pi}{8}\) & \(2.51\times 10^{-10}\) & \(2.51\times 10^{-10}\) & \(2.51\times 10^{-10}\) & \(2.51\times 10^{-10}\) \\ \(\frac{\pi}{2}\) & \(2.51\times 10^{-10}\) & \(2.51\times 10^{-10}\) & \(2.51\times 10^{-10}\) & \(2.51\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculated error values for CD schemes with equal \((M+N)\).