We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of \(L^2\) compared to classical results in dimension d, which are limited to higher-order (sufficiently smooth) Sobolev spaces \(H^s\) with \(s>d/2\). In particular, we are able to establish a global error estimate in \(L^2\) for \(H^1\) solutions which is roughly of order \(\tau ^{ {1\over 2} + { 5-d \over 12} }\) in dimension \(d \le 3\) (\(\tau \) denoting the time discretization parameter). This breaks the “natural order barrier” of \(\tau ^{1/2}\) for \(H^1\) solutions which holds for classical numerical schemes (even in combination with suitable filter functions).

我们基于最近的时间离散化和滤波技术，为立方非线性Schrödinger方程提供了一种新的经滤波的低正则傅立叶积分器。对于这种新方案，我们进行了严格的误差分析，并以低规则性建立了比迄今为止文献中的经典方案更好的收敛速度。在我们的误差估计中，我们将新方案的更好的局部误差特性与基于一般离散Strichartz类型估计的稳定性分析相结合。后者使我们能够处理更粗糙的解决方案，因为与维d中的经典结果相比，误差分析可以直接在\（L ^ 2 \）级别进行，而维度d仅限于高阶（足够平滑） ）Sobolev空间\（H ^ s \）与\（s> d / 2 \）。特别是，我们能够为\（H ^ 1 \）解决方案在\（L ^ 2 \）中建立全局误差估计，该估计大约是\（\ tau ^ {{1 \ over 2} + {5- d \ over 12}} \）维度\（d \ le 3 \）（\（\ tau \）表示时间离散化参数）。这打破了\（\ tau ^ {1/2} \）对于\（H ^ 1 \）解的“自然阶障碍”，该解适用于经典数值方案（甚至与适当的过滤函数结合使用）。