In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with sharp dependence on the regularity index of the solution from a wide range that is approaching the lower bound for well-posedness. Particularly for very rough data, the technique of discrete Bourgain space is utilized and developed, which can apply for general second-order wave models. Numerical verifications are provided.

翻译：本文研究非线性Klein--Gordon/波动方程在粗糙初值条件下时间分裂格式的收敛性分析。针对Lie分裂格式与Strang分裂格式，我们建立了其误差估计的最优界，该界对解的正则性指标具有尖锐依赖性，且所考虑的正则性范围广泛，已接近适定性理论所要求的下界。特别针对极粗糙数据，我们运用并发展了离散Bourgain空间技术，该方法可推广至一般二阶波动模型。文中同时提供了数值验证。