We propose and analyze a novel symmetric exponential wave integrator (sEWI) for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form $ f(\rho) = \rho^\sigma $, where $ \rho:=|\psi|^2 $ is the density with $ \psi $ the wave function and $ \sigma > 0 $ is the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For "good" potential and nonlinearity ($H^2$-potential and $\sigma \geq 1$), we establish an optimal second-order error bound in $L^2$-norm. For low regularity potential and nonlinearity ($L^\infty$-potential and $\sigma > 0$), we obtain a first-order $L^2$-norm error bound accompanied with a uniform $H^2$-norm bound of the numerical solution. Moreover, adopting a new technique of \textit{regularity compensation oscillation} (RCO) to analyze error cancellation, for some non-resonant time steps, the optimal second-order $L^2$-norm error bound is proved under a weaker assumption on the nonlinearity: $\sigma \geq 1/2$. For all the cases, we also present corresponding fractional order error bounds in $H^1$-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.

翻译：本文针对具有低正则势和典型幂律非线性项$ f(\rho) = \rho^\sigma $（其中$ \rho:=|\psi|^2 $为波函数$ \psi $的密度，$ \sigma > 0 $为非线性指数）的非线性薛定谔方程（NLSE），提出并分析了一种新型对称指数波积分器（sEWI）。该格式为显式格式，在时间步长限制下保持稳定性，且该限制与网格尺寸无关。我们在势函数和非线性项的不同正则性假设下，严格建立了sEWI的误差估计。对于“良好”情形（$H^2$正则势且$ \sigma \geq 1 $），我们在$L^2$范数下建立了最优的二阶误差界。对于低正则情形（$L^\infty$正则势且$ \sigma > 0 $），我们在数值解均匀$H^2$范数有界的条件下，得到了$L^2$范数的一阶误差界。进一步地，通过引入新的“正则补偿振荡”（RCO）技术分析误差抵消，对于某些非共振时间步，我们在非线性项较弱假设（$ \sigma \geq 1/2 $）下证明了最优的二阶$L^2$范数误差界。针对所有情形，我们还给出了相应的$H^1$范数分数阶误差界，该范数是能量意义下的自然范数。大量数值结果验证了我们的误差估计，并展示了sEWI相较于现有方法的优越性，包括对势函数和非线性项极低的正则性要求，以及近乎守恒质量和能量的优异长时间行为。