论文标题
以低规律性以低规律性求解的半差异方案的一类半差异方案的错误分析
Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity
论文作者
论文摘要
我们分析了一类时间离散化，用于求解具有非平滑电位的非线性schrödinger方程，并且在任意Lipschitz域上的低规度和低规范性$ω\ subset \ subset \ mathbb {r}^d $，$ d $，$ d \ le 3 $ 3 $。我们表明，这些方案及其最佳局部误差结构允许在解决方案和电势较低的规律性假设下进行收敛，而不是经典方法（例如分裂或指数积分方法）所要求的。此外，我们在周期性的边界条件下显示了第一阶和二阶收敛，在任何分数的sobolev space $ h^{r} $，$ r \ ge 0 $中，超出了更典型的$ l^2 $或$ h^σ（σ> \ frac {d} {d} {2} {2} {2} $）。数值实验说明了我们的结果。
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.
