finite difference method for the Korteweg-de Vries-Kawahara equation

Abstract

In this work, we aim to study numerical solutions of a nonlinear partial differential equation using a family of three-level linearized finite difference $\theta$-methods for a shallow water waves having surface tension in the form of Viscous Korteweg-de Vries-Kawahara equation with initial and boundary conditions. The model admits two invariants: momentum and energy. The scheme is proved to preserve both momentum and energy in the discrete sense when θ = 1/3. In addition, we proved that the method converges uniformly. The method gives second-order of accuracy in space. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.