A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

Abstract

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

“…To give some examples, by utilizing Lie group iterative scheme, Chang and Liu [44] solved nonlinear Klein-Gordon and sine-Gordon equations, a Lie group scheme combined with RBFs is applied to Burgers equation by Seydaoglu [45], a composition of reproducing kernel method and GPS is used for Lane-Emden equation [46], some heat conduction equations are investigated by Liu [47], Liu [48] solved Burgers equation by using finite difference and group preserving scheme (GPS), numerical solutions of some PDEs are obtained by Hajiketabi et al [49,50] utilizing meshless methods combined with geometric numerical integrators, and Klein-Gordon equation is considered by Gao et al [51]. Recently, KdV equation is solved by Polat and Oruç [52] via combination of Lie group scheme and DBFs, and accurate results are obtained. Furthermore, exponential Lie group scheme is proposed for Burgers equation by Seydaoglu [53].…”

Computational and numerical wave solutions of the Caudrey–Dodd–Gibbon equation

“…The fifth–order KdV (fKdV) equation which is the most well-known model of shallow water wave propagation, is given by [26] where are arbitrary positive parameters [27] . Moreover, has various practical applications in a wide range of fields, including quantum mechanics and nonlinear optics, and it shows how long waves flow in shallow water under gravity and in a one-dimensional nonlinear lattice.…”