We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order, both on the torus $\mathbb{T}^d$ and on a smooth bounded domain $\Omega \subset \mathbb{R}^d$, $d\le 3$, equipped with homogeneous Dirichlet boundary condition. The new scheme allows for a symmetric approximation to the NLS equation in a more general setting than classical splitting, exponential integrators, and low-regularity schemes (i.e. under lower regularity assumptions, on more general domains, and with fractional rates). We motivate and illustrate our findings through numerical experiments, where we witness better structure preserving properties and an improved error-constant in low-regularity regimes.

翻译：本文提出并分析了一种超越经典傅里叶技术的非线性薛定谔方程对称低正则性格式。我们证明了该格式在$L^2$范数下具有从一阶到二阶的分数阶收敛性，该收敛性在环面$\mathbb{T}^d$和具有齐次狄利克雷边界条件的光滑有界域$\Omega \subset \mathbb{R}^d$（$d\le 3$）上均成立。相比于经典分裂法、指数积分器和低正则性格式，新格式在更一般的设定（即更低的正则性假设、更一般的域以及分数阶收敛速率）下实现了对非线性薛定谔方程的对称逼近。我们通过数值实验验证并阐释了相关发现，观察到在低正则性条件下，该格式具有更好的结构保持特性及更优的误差常数。