Collab. Research: Instability analysis of the split-step method on spatially-varying backgrounds, with applications to optical telecommunications and Bose-Einstein condensation

算子分裂法（operator-splitting，简称分裂步法）被广泛用于数值求解从流体力学到量子力学的各种应用中产生的含时偏微分方程。为了最小化计算时间，需要选择尽可能大的时间步长。另一方面，时间步长的上界通常由数值格式稳定的要求来设定。冯诺依曼分析是用来获得这样的上限模型的系数是常数的问题。然而，实际上感兴趣的方程的解通常在空间中不是常数。为了证明使用冯诺依曼分析这样的问题，人们往往近似的非常数系数的常数。然而，对于SSM，这种方法失败了。最近，我们提出了一种替代方法来分析SSM的不稳定性，当这种方法用于模拟接近孤子的解（即，钟形解）的非线性薛定谔方程。在这个项目中，我们将扩展分析，以更实际的相关设置，涉及两个应用：光纤通信和玻色？爱因斯坦凝聚。这将提供一个基本上是非常数系数的问题的数值不稳定性的发展的理解。然后，我们将使用这些信息提出修改的SSM放宽稳定性要求。显然，这将减少计算时间。这个项目将开发一个系统的方法来研究一个广泛使用的数值方法，SSM的基本属性“稳定性”。数值方法必须是稳定的，以便准确地模拟感兴趣的物理过程。目前的稳定性分析方法是用一些常数值来近似模拟过程。我们不会使用这种近似，因为我们已经证明，它会导致对SSM性能的错误预测。我们的替代方法将依赖于数值分析和线性微分方程理论的技术组合。它将提供对SSM的性能限制的理解。反过来，这将使我们能够提出更有效和可靠的修改这种数值方法。在这个项目中考虑的应用将直接影响光纤通信系统和低温原子凝聚体的建模。然而，我们的方法将影响其他应用的SSM，其中包括环境建模，水文，热传导和反应流。此外，该方法可以扩展到相关的数值方法，这些方法用于其他应用中，例如通过反应和随机运动（扩散）对分子和化学物质之间的相互作用进行建模。

The operator-splitting, or split-step, method (SSM) is widely used to numerically solve time-dependent partial differential equations arising in diverse applications, from hydrodynamics to quantum mechanics. To minimize the computational time, one needs to select the time step as large as possible. On the other hand, the upper bound on the time step is often set by the requirement that the numerical scheme be stable. The von Neumann analysis is used to obtain such upper bounds for model problems where the coefficients are constant. However, solutions of practically interesting equations are typically not constant in space. To justify the use of the von Neumann analysis for such problems, one often approximates non-constant coefficients by constant ones. However, for the SSM, this approach fails. Recently, we proposed an alternative approach to analyze the instability of the SSM when this method is used to simulate a solution close to the soliton (i.e., a bell-shaped solution) of the nonlinear Schroedinger equation. In this project, we will extend that analysis to more practically relevant settings that involve two applications: fiber optical telecommunications and Bose?Einstein condensates. This will provide an understanding of the development of the numerical instability in problems with essentially non-constant coefficients. We will then use this information to propose modifications of the SSM with relaxed stability requirements. Clearly, this will reduce the computational time.This project will develop a systematic approach to studying a fundamental property "stability" of a widely used numerical method, the SSM. A numerical method must be stable in order to accurately model the physical process of interest. The current approach to the stability analysis consists in approximating the simulated processes by some constant values. We will not use this approximation, as we have demonstrated that it leads to incorrect predictions regarding the performance of the SSM. Our alternative approach will rely on a combination of techniques from numerical analysis and the theory of linear differential equations. It will provide an understanding of the performance limitations of the SSM. This, in turn, will allow us to propose more efficient and reliable modifications of this numerical method. The applications considered in this project will directly impact the modeling of fiber-optic communication systems and low-temperature atomic condensates. However, our approach will affect other applications of the SSM, which include environmental modeling, hydrology, heat conduction, and reacting flows. Moreover, the approach can be extended to related numerical methods, which are used in other applications such as the modeling of the interaction among molecules and chemical species through reactions and random motion (diffusion).