低正则性下求解Gross-Pitaevskii方程的一类半离散格式的误差分析
Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity
摘要：
本文研究了任意Lipschitz区域$Ω\subset \mathbb{R}^d$，$d \le 3$上具有非光滑势和低正则性的非线性薛定谔方程的一类时间离散方法.我们表明，这些计划，连同其最佳的局部误差结构，允许收敛在较低的正则性假设下的解决方案和潜在的比所需的经典方法，如分裂或指数积分方法。此外，我们显示了一阶和二阶收敛的情况下，周期边界条件，在任何分数正Sobolev空间$H^{r}$，$r \ge 0$，超越了更典型的$L^2$或$H^σ(σ>\frac{d}{2}$）-误差分析。数值实验说明了我们的结果。
英文摘要：
We analyse a class of time discretizations for solving the nonlinear Schrödinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $Ω\subset \mathbb{R}^d$, $d \le 3$. 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 $H^σ(σ>\frac{d}{2}$) -error analysis. Numerical experiments illustrate our results.
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