AI 中文总结
本研究针对DSW形成前无色散极限区的gKdV方程，证明CN与Lawson-RK格式在傅里叶伪谱离散下具有ε一致的二阶时间精度与谱空间精度，还验证了CN格式的唯一可解性及步长限制的必要性。
AI 中文摘要
本研究针对广义Korteweg-de Vries（gKdV）方程在无色散极限区（具体为色散激波（DSW）形成前）应用经典数值格式时的一致误差估计问题展开研究。我们分析了Crank-Nicolson（CN）格式与一种Lawson型Runge-Kutta（Lawson-RK）格式，二者均通过傅里叶伪谱法进行空间离散。我们证明，两种全离散格式均达到最优二阶时间精度与谱空间精度，且误差常数在消失色散参数ε下保持一致。该分析还解决了无色散极限区中CN格式的唯一可解性问题。理论结果得到数值实验支持，实验验证了ε一致精度，同时表明CN格式需对步长施加限制的必要性。本研究总体上验证了求解DSW形成前无色散gKdV方程的经典方法的有效性。
英文摘要
This work establishes uniform error estimates for classical numerical schemes applied to the generalized Korteweg-de Vries (gKdV) equation in the dispersionless limit regime, specifically before the development of dispersive shock wave (DSW). We analyze the Crank-Nicolson (CN) and a Lawson-type Runge-Kutta (Lawson-RK) methods, when discretized in space via the Fourier pseudo-spectral method. We prove that both fully discrete schemes achieve optimal second-order temporal accuracy and spectral spatial accuracy, with error constants being uniform in the vanishing dispersion parameter $\varepsilon$. The analysis also addresses the unique solvability of the CN scheme in the dispersionless limit regime. Theoretical findings are supported by numerical experiments, demonstrating the $\varepsilon$-uniform accuracy and the necessity of step size restriction for CN. The study in general validates the classical methods for solving the dispersionless gKdV equation before DSW.
