We investigate a filtered Lie-Trotter splitting scheme for the ``good" Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in $H^{s}$ for $0<s\leq 2$, overcoming the constraint of $s>1/2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order $\tau^{s/2}$ in $L^2$ for such levels of regularity. Our analytical findings are supported by numerical experiments.

翻译：本文研究了一种用于求解“好”Boussinesq方程的滤波Lie-Trotter分裂格式，并针对极低正则性的初始数据推导了误差估计。通过使用离散Bourgain空间，我们的分析将初始数据的正则性要求扩展至 $H^{s}$ 空间，其中 $0<s\leq 2$，从而克服了光滑Sobolev空间中双线性估计所要求的 $s>1/2$ 的限制。我们证明了在此正则性水平下，该格式在 $L^2$ 范数下具有 $\tau^{s/2}$ 阶的收敛速率。数值实验支持了我们的理论分析结果。