Abstract In this paper, the nonlinear dispersion wave model in both 1D and 2D is studied by the compact finite difference method, which is called the generalized Rosenau-RLW equation. A fourth-order compact three-level and linearized difference scheme that maintains the original conservative properties of equation is proposed. The discrete mass conservation and discrete energy conservation of compact difference scheme are obtained. The solvability of numerical scheme is obtained. By using the discrete energy method, convergence and unconditional stability can also be obtained without relying on the grid ratio, and the optimal error estimates in the L ∞ norm are fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. The scheme is conservative so can be used for long time computation. Numerical experiment results show that the theory is accurate and the method is efficient and reliable. Finally, the new numerical scheme is used to study the nonlinear dynamic of 1D generalized Rosenau-RLW equation and the wave interference of 2D generalized Rosenau-RLW equation, focusing on the influence of nonlinear convection term on the wave and the conservation of wave propagation.

中文翻译：

---

基于四阶紧保守差分格式的一维和二维非线性色散模型动力学研究

摘要 本文采用紧致有限差分法研究了一维和二维非线性色散波模型，称为广义Rosenau-RLW方程。提出了一种保持方程原有保守性质的四阶紧致三级线性差分格式。得到了紧差分格式的离散质量守恒和离散能量守恒。获得了数值方案的可解性。通过使用离散能量方法，也可以在不依赖网格比的情况下获得收敛性和无条件稳定性，L ∞ 范数中的最优误差估计分别为空间和时间步长的四阶和二阶精度. 该方案是保守的，因此可以用于长时间计算。数值实验结果表明，该理论是准确的，方法是有效可靠的。最后，利用新的数值方案研究一维广义Rosenau-RLW方程的非线性动力学和二维广义Rosenau-RLW方程的波干涉，重点研究非线性对流项对波的影响和波传播守恒。