## 1 Introduction  
High-order numerical methods are commonly used in solving mathematical and engineering problems described by the class of higher-order partial differential equations. They are used, among others, in modeling wave propagation (second, third, fourth, fifth, and higher orders), combustion (third, fourth, and fifth orders), thermal instability (sixth and eighth orders), magnetohydrodynamics, or astrophysics (tenth and twelfth orders). In computational fluid dynamics, the higher-order derivatives, e.g., of fourth or sixth orders, are used to introduce artificial viscosity terms or to approximate filter functions [1]. An accurate approximation of the higher-order derivatives in these problems requires the application of sophisticated and highly accurate methods as, otherwise, the induced errors act as artificially introduced terms destroying the solution accuracy. Various numerical methods used for higher-order derivative discretization were presented and analyzed in a review paper by [PERSON] and [PERSON] [2]. In the present paper, we focus on compact difference (CD) schemes of very high orders, e.g., 20 th or 30 th, and demonstrate their accuracy based on canonical test cases. The approximation of derivative by CD schemes relies on solving a linear system of equations formulated for all grid points along a given node line. They can be viewed as a generalization of the well-known finite difference (FD) methods. CD schemes are used for both incompressible and compressible flow analysis [3; 4; 5; 6], including turbulence [7; 8], aeroacoustics [9; 10; 11], and combustion phenomena [12; 13; 14]. In particular, the CD schemes have found wide applications in the problem of propagation of acoustic [15; 16; 17] and electromagnetic waves [18; 19]. They provide accurate results over long periods of time and over long spatial domains. In this paper, we deal with high-order partial differential equations (PDEs) describing linear and non-linear dispersion waves whose phase and group velocities depend on the frequency. In non-linear problems of dispersive waves, the solutions in the form of an infinitely long, periodic wave packet are well known. One example is the so-called cnoidal wave in the theory of water waves, which is a non-linear and exact periodic wave solution of the Korteweg-de Vries equation [20]. Korteweg-de Vries (KdV)-like equations are non-linear PDEs of the third, fifth, seventh, or higher orders. The standard third-order KdV equation was introduced by [PERSON] in 1877 [21] and then rediscovered by [PERSON] and [PERSON] in 1895 [20]. The solutions to the KdV equations are solitary waves, known as solitons stemming from the cancellation of non-linear and dispersive effects. Their characteristic feature is that they preserve shape during propagation. The KdV-like equations are used to model non-linear dispersive waves in a wide range of applications such as fluid dynamics (e.g., shallow-water waves [20; 22; 23; 24], gravity-capillary waves [25; 26]), optics (optical fibers [27]), plasma physics (magneto-acoustic waves [28; 29], ion-acoustic waves [30; 31]), and quantum mechanics [32; 33]. The numerical methods commonly used to approximate KdV equations include finite difference methods, spectral collection methods, and Galerkin methods. [PERSON] et al. [34] applied both FD and CD schemes for KdV problems involving the third-order derivative. They compared the results with a previous study by [PERSON] and [PERSON] [35] who used a high-order Local Discontinuous Galerkin (LDG) method, and a study by [PERSON] et al. [36], based on the adaptive line method. [PERSON] et al. [37] used FD schemes based on a predictor-corrector algorithm and a linearized implicit corrector scheme for the third and fifth-order KdV equations. [PERSON] [38] also applied FD and CD schemes for differential equations containing fourth and fifth-order derivatives in a conjunction with a low-pass filter. The results were also compared with the LDG method applied by [PERSON] and [PERSON] [39].