We analyse a class of time discretizations for solving the nonlinear Schr\"odinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \le 3$. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space $H^{r}$, $r \ge 0$, beyond the more typical $L^2$ or $H^\sigma (\sigma>\frac{d}{2}$) -error analysis. Numerical experiments illustrate our results.
翻译:我们分析的是用于解决非线性Schr\'odinger等方程式的不线性分解类别,该方程式具有非线性潜力,且在任意的Lipschitz域中以低常规形式($\Omega\subset\mathb{R ⁇ d$,$d\le$3美元)解决非线性Schr\'odinger等方程式。我们发现,这些计划及其最佳的本地误差结构,在较低的常规假设下,既可以在解决方案上趋同,也可以在传统方法(如分裂或指数化集成法)要求的潜力上趋同。此外,在定期边界条件下,在任何分数正数的Sobolev空间($H ⁇ r},$r\ge 0,$r\ge$,超过典型的$L ⁇ 2美元或$H>sgma(\gma{d{d ⁇ 2$)-error 分析中,我们可以看到第一和第二顺序的趋同。数字实验显示了我们的结果。