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：我们提出并分析了一种超越经典傅里叶方法的非线性薛定谔方程（NLS）对称低正则性格式。我们证明了该格式在$L^2$-范数下，在环面$\mathbb{T}^d$以及光滑有界区域$\Omega \subset \mathbb{R}^d$（$d\le 3$，配备齐次狄利克雷边界条件）上具有从一阶到二阶的分步收敛性。新格式能够在比经典分裂法、指数积分器和低正则性格式（即在更低正则性假设、更一般的区域以及分步率条件下）更广泛的框架下实现NLS方程的对称近似。我们通过数值实验验证并阐释了研究发现，观察到在低正则性区域中更好的结构保持性质和改进的误差常数。