We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schr\"odinger and wave equations under the assumption of $H^1$ solutions. For the Schr\"odinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schr\"odinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates.

翻译：我们证明了在$H^1$解假设下，特定低正则性积分器应用于一维周期非线性薛定谔方程与波动方程时的最优收敛速率。对于薛定谔方程，我们分析了Ostermann与Schratz于2018年提出的指数型格式；对于波动方程，我们处理了Li、Schratz与Zivcovich于2023年提出的修正李分裂法。我们证明这些积分器分别能以完整的一阶和二阶精度收敛，而此前在此类情形下仅知分数阶收敛速率。证明的关键要素是对应线性问题解的已知时空界限：在薛定谔方程情形中采用$L^4$ Strichartz不等式，在波动方程情形中采用零形式估计。据我们所知，这是零形式估计首次在数值分析领域得到应用。我们采用连续时间框架下的估计，从而避免了离散时间估计可能导致的精度损失。