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.

本文的主要贡献是介绍了新的紧致四阶，标准四阶和标准二阶有限差分方案，用于求解Kawahara方程，五阶偏导数方程。证明了通过紧致的四阶有限差分格式获得的数值解的仅质量守恒。但是，标准的四阶和标准的二阶有限差分方案可以同时保留质量和能量。冯·诺依曼分析也证明了稳定性。通过对数值实验的分析，给出了每种方案的精度顺序和紧凑方案的计算效率。为了验证所提出方法的潜力，我们还考虑了长期行为。最后，