Performance of compact and non-compact structure preserving algorithms to traveling wave solutions modeled by the Kawahara equation

The main contribution of this article is to introduce new compact fourth-order, standard fourth-order, and standard second-order finite difference schemes for solving the Kawahara equation, the fifth-order partial derivative equation. The conservation of mass only of the numerical solution obtained by the compact fourth-order finite difference scheme is proven. However, the standard fourth-order and standard second-order finite difference schemes can preserve both mass and energy. The stability is also proven by von Neumann analysis. According to analysis for numerical experiments, the order of accuracy for each scheme and the computational efficiency of the compact scheme are presented. To validate the potential of the presented methods, we also consider long-time behavior. Finally, results obtained from the compact scheme are superior than those from the non-compact schemes.