An efficient fourth-order three-point scheme for solving some nonlinear dispersive wave equations

Abstract

In this work, based on the value computed by the fourth-order implicit Padé scheme for the first derivative, a fourth-order three-point compact scheme for approximating the third derivative is suggested and applied to numerically solve some nonlinear dispersive wave equations including the Korteweg de Vires (KdV) equation and the two-dimensional Zakharov–Kuznetsov (ZK) equation. Spectral properties of the proposed scheme are analyzed and compared with other high-order schemes, and it is found that the proposed scheme has satisfactory numerical properties similar to high-order compact schemes. An optimized boundary closure scheme for the approximation of the first derivative is proposed to improve numerical stability. Numerical results of KdV equation and ZK equation obtained by the present scheme match very well with analytic solutions and available numerical results and showcase excellent performance of the scheme to solve nonlinear problems. And owing to the explicit formula for the second and the third derivative, the present scheme is efficient and consumes noticeable less computation time for solving the KdV equation compared to existing three-point high-order compact schemes.