带耗散项RLW方程的高阶有限差分算法

High-Order Finite Difference Algorithm for RLW Equation with Dissipation Term

摘要: 为了利用非线性RLW方程的初边值问题进行数值模拟研究，首先构造高阶守恒型紧致有限差分格式，对时间方向的导数采用隐中点格式进行离散，对空间方向的各阶导数用逆紧致算子进行离散，使其在时间方向上的精度达到二阶，在空间方向上的精度达到六阶。进一步，对差分格式进行守恒性证明、先验估计及收敛性分析。最后，通过数值算例验证理论的正确性和格式的有效性与可靠性。

Abstract: To conduct numerical simulation research on the initial-boundary value problem of the nonlinear RLW equation, a high-order conservative compact finite difference scheme is first constructed. The derivative in the time direction is discretized using the implicit mid-point scheme, and the derivatives of each order in the spatial direction are discretized using the inverse compact operator, achieving second-order accuracy in the time direction and sixth-order accuracy in the spatial direction. Further, the conservation property of the difference scheme is proved, a priori estimates are made, and the convergence analysis is carried out. Finally, the correctness of the theory and the effectiveness and reliability of the scheme are verified through numerical examples.