带$\mathbb{R}^d$上势场的薛定谔方程移动窗口法的收敛性
Convergence of a moving window method for the Schrödinger equation with potential on $\mathbb{R}^d$
AI总结：
针对带外势的全空间薛定谔方程，提出随解移动的有界标度环面窗口截断框架，结合已有周期格式构造全空间算法，严格证明了时空收敛阶，并通过数值实验验证了理论与方法的有效性。
摘要：
我们提出了一种名为移动窗口法的新框架，用于求解$\mathbb{R}^d$中带外势的线性薛定谔方程。该方法采用光滑截断函数，将全空间柯西边界条件下的方程截断至一个随解移动的标度环面有界窗口内。由此便可利用该标度环面上的已有格式来设计全空间问题的算法。本文严格分析了有界窗口上的数值解逼近全空间解时的误差，此外还借助周期情形的分析工具，严格估计了这些全空间算法的误差。将所提框架与标度环面上的经典一阶指数积分器相结合，我们证明了对于满足$γ\geq 2$的$H^γ(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2γ} dx)$类初值，该格式在时间上达到一阶收敛，在空间上达到$γ/2$阶收敛。在$γ= 1$的情形下，数值格式在附加CFL条件下具有半阶收敛性。在实际应用中，当波到达窗口边界时，我们可以动态调整窗口，从而在初始窗口之外继续计算。本文给出了大量数值算例，以支撑理论分析并验证所提方法的有效性。
英文摘要：
We propose a novel framework, called moving window method, for solving the linear Schrödinger equation with an external potential in $\mathbb{R}^d$. This method employs a smooth cut-off function to truncate the equation from Cauchy boundary conditions in the whole space to a bounded window of scaled torus, which is itself moving with the solution. This allows for the application of established schemes on this scaled torus to design algorithms for the whole-space problem. Rigorous analysis of the error in approximating the whole-space solution by numerical solutions on a bounded window is established. Additionally, analytical tools for periodic cases are used to rigorously estimate the error of these whole-space algorithms. By integrating the proposed framework with a classical first-order exponential integrator on the scaled torus, we demonstrate that the proposed scheme achieves first-order convergence in time and $γ/2$-order convergence in space for initial data in $H^γ(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2γ} dx)$ with $γ\geq 2$. In the case where $γ= 1$, the numerical scheme is shown to have half-order convergence under an additional CFL condition. In practice, we can dynamically adjust the window when waves reach its boundary, allowing for continued computation beyond the initial window. Extensive numerical examples are presented to support the theoretical analysis and demonstrate the effectiveness of the proposed method.
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