We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szeg\H{o} equations on the torus, which are $\textit{exact in time}$ with $\textit{spectral accuracy}$ in space. We prove spectral convergence for the first two equations, of order $K^{-s+1}$ for initial data in $H^s(\mathbb T)$, with an error constant depending $\textit{linearly}$ on the final time instead of exponentially. These schemes are based on $\textit{explicit formulas}$, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales. These schemes open doors for the understanding of the long-time dynamics of integrable equations.

翻译：本文针对环面上的Benjamin-Ono方程、Calogero-Sutherland DNLS方程及三次Szeg\H{o}方程，构建了具有$\textit{时间精确性}$与$\textit{空间谱精度}$的全离散格式。对于前两个方程，我们证明了其谱收敛性：当初始数据属于$H^s(\mathbb T)$空间时，收敛阶为$K^{-s+1}$，且误差常数随终止时间呈$\textit{线性}$增长而非指数增长。这些格式建立在$\textit{显式公式}$基础上，此类公式近年来在非线性可积方程理论中逐渐涌现。数值模拟表明，新构建的方法在短时间与长时间尺度上均表现出优越性能。这些格式为理解可积方程的长时间动力学行为开辟了新途径。