We establish optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and typical power-type nonlinearity $f(\rho )={\rho }^{\sigma }$, where $\rho :=|\psi {|}^2$ is the density with $\psi$ the wave function and $\sigma >0$ the exponent of the nonlinearity. For the first-order Lie–Trotter time-splitting method, optimal $L^2$-norm error bound is proved for $L^{\infty }$-potential and $\sigma >0$, and optimal $H^1$-norm error bound is obtained for $W^{1,4}$-potential and $\sigma \ge 1/2$. For the second-order Strang time-splitting method, optimal $L^2$-norm error bound is established for $H^2$-potential and $\sigma \ge 1$, and optimal $H^1$-norm error bound is proved for $H^3$-potential and $\sigma \ge 3/2$ (or $\sigma =1$). Compared to those error estimates of time-splitting methods in the literature, our optimal error bounds either improve the convergence rates under the same regularity assumptions or significantly relax the regularity requirements on potential and nonlinearity for optimal convergence orders. A key ingredient in our proof is to adopt a new technique called regularity compensation oscillation (RCO), where low frequency modes are analyzed by phase cancellation, and high frequency modes are estimated by regularity of the solution. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

我们为具有低正则势和典型幂型非线性的非线性薛定谔方程的时间分割方法建立了最佳误差界$F（\rho ）={\rho }^{\sigma }$， 在哪里$\rho ：=|\psi {|}^2$是密度$\psi$波函数和$\sigma >0$非线性指数。对于一阶 Lie–Trotter 时间分割方法，最优$L^2$- 范数误差界被证明为$L^{\mathrm{无穷大}}$-潜力和$\sigma >0$，和最优的$H^1$- 获得范数误差界${瓦}^{1,4}$-潜力和$\sigma \ge 1/2$。对于二阶Strang时间分割方法，最优$L^2$- 建立范数误差界限$H^2$-潜力和$\sigma \ge 1$，和最优的$H^1$- 范数误差界被证明为$H^3$-潜力和$\sigma \ge 3/2$（或者$\sigma =1$）。与文献中时间分割方法的误差估计相比，我们的最优误差界要么提高了相同正则性假设下的收敛速度，要么显着放宽了对最优收敛阶数的势和非线性的正则性要求。我们证明的一个关键要素是采用一种称为规律性补偿振荡（RCO）的新技术，其中通过相位抵消来分析低频模式，并通过解的规律性来估计高频模式。报告了广泛的数值结果，以证实我们的错误估计并证明它们是敏锐的。