We study time integration schemes for $\dot H^1$-solutions to the energy-(sub)critical semilinear wave equation on $\mathbb{R}^3$. We show first-order convergence in $L^2$ for the Lie splitting and convergence order $3/2$ for a corrected Lie splitting. To our knowledge this includes the first error analysis performed for scaling-critical dispersive problems. Our approach is based on discrete-time Strichartz estimates, including one (with a logarithmic correction) for the case of the forbidden endpoint. Our schemes and the Strichartz estimates contain frequency cut-offs.

翻译：我们研究了能量（次）临界半线性波动方程在$\mathbb{R}^3$上的$\dot H^1$解的时间积分格式。对于Lie分裂，我们证明了$L^2$中的一阶收敛性，而对于修正的Lie分裂，则证明了收敛阶为$3/2$。据我们所知，这包括首次针对尺度临界色散问题进行的误差分析。我们的方法基于离散时间Strichartz估计，包括一个（带有对数修正）针对禁止端点情形的估计。我们的格式和Strichartz估计包含频率截断。