We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the long-time error bounds of different low-regularity integrators for the nonlinear Schr\"odinger equation (NLSE) with small initial data characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in $H^r$ is proven at $O(\varepsilon \tau)$ for solutions in $H^r$ with $r > 1/2$ up to the time $T_{\varepsilon } = T/\varepsilon $ with fixed $T > 0$. Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time $T_{\varepsilon } = T/\varepsilon ^2$. With the help of the regularity compensation oscillation (RCO) technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduce the long-time error by a factor of $\varepsilon^2$ compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.
翻译：对于非线性Schr\'odinger方程式(NLSE),我们引入了一个新的非共振低规则整合器(NLSE),允许进行长期误差估算,这种估算在底部PDE的意义上是最佳的。因此,主要理念在于完全在离散范围内处理零模式。对于长期误差估算,我们严格地为非线性Schr\'odinger方程式(NLSE)设置了不同低规则整合器的长期误差界限。对于非线性Schr\'od 方程式(NLSE),最初数据以无维度参数为特征 $(0,1,1美元 准确度),长期误差值(NRC-l),先是平流性 NRC-deal-rder-lational-lationalislational dislental ral-rational-lational-lentral-lational-rental-ral-ral-ral-lational-ral-lational-rational-rational-ral-ral-ral-ral-lational-ral-ral-ral-lation-lation-ral-ral-ral-ral-ral-ral-ral-lxxxxxxxxx-lxxxxxxxxx-l-lxxx-lx-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-
相关内容
微信扫码咨询专知VIP会员