用稳定高效的数值方法模拟浅水波演化，精度显著优于传统方法。
Using Crank-Nikolson Scheme to Solve the Korteweg-de Vries (KdV) Equation
- 采用隐式差分的Crank-Nicolson方案求解KdV方程
- 数值结果表明方法收敛性好，误差更小
- 适合需要高精度模拟非线性波动的物理与工程研究
Korteweg-de Vries（KdV）方程是描述浅水波及其他色散介质中波传播的基本偏微分方程。准确求解KdV方程对理解物理与工程中的波动力学至关重要。本项目实现了一种以稳定性与精度著称的有限差分方法——Crank-Nicolson方案来求解KdV方程。该方案的隐式特性使其在处理KdV方程中的色散项和非线性项时具有更好的数值稳定性。通过多个测试案例评估了该方法的性能，分析其收敛性与误差行为。结果表明，Crank-Nicolson方法为求解KdV方程提供了一种稳健且精确的途径，相比传统显式方法具有更高精度。代码可在论文末尾获取。
原文摘要 · Abstract (English)
The Korteweg-de Vries (KdV) equation is a fundamental partial differential equation that models wave propagation in shallow water and other dispersive media. Accurately solving the KdV equation is essential for understanding wave dynamics in physics and engineering applications. This project focuses on implementing the Crank-Nicolson scheme, a finite difference method known for its stability and accuracy, to solve the KdV equation. The Crank-Nicolson scheme's implicit nature allows for a more stable numerical solution, especially in handling the dispersive and nonlinear terms of the KdV equation. We investigate the performance of the scheme through various test cases, analyzing its convergence and error behavior. The results demonstrate that the Crank-Nicolson method provides a robust approach for solving the KdV equation, with improved accuracy over traditional explicit methods. Code is available at the end of the paper.
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