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

算子分裂或分步方法（SSM）广泛用于数值求解从流体力学到量子力学等各种应用中出现的时变偏微分方程。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是由数值格式稳定的要求来设定的。对于系数为常数的模型问题，采用冯·诺伊曼分析来获得这种上界。然而，实际有趣的方程的解在空间中通常不是常数。为了证明对这类问题使用冯·诺依曼分析是正确的，人们常常用常数来近似非常数系数。然而，对于SSM来说，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，当该方法用于模拟非线性薛定谔方程的接近孤子的解（即钟形解）时。在这个项目中，我们将把分析扩展到更实际的相关设置，涉及两种应用：光纤通信和Bose？爱因斯坦冷凝物。这将有助于理解非常系数问题中数值不稳定性的发展。然后，我们将利用这些信息提出修改SSM的建议，放宽稳定性要求。显然，这将减少计算时间。本项目将开发一种系统的方法来研究广泛使用的数值方法SSM的基本性质“稳定性”。为了准确地模拟所关注的物理过程，数值方法必须是稳定的。目前的稳定性分析方法是用一些常数来近似模拟过程。我们不会使用这种近似，因为我们已经证明，它会导致关于SSM性能的错误预测。我们的替代方法将依赖于数值分析技术和线性微分方程理论的结合。它将提供对SSM性能限制的理解。反过来，这将使我们能够对这种数值方法提出更有效和可靠的修改。本项目所考虑的应用将直接影响光纤通信系统和低温原子凝聚物的建模。然而，我们的方法将影响SSM的其他应用，包括环境建模、水文学、热传导和反应流。此外，该方法可以扩展到相关的数值方法，用于其他应用，如通过反应和随机运动（扩散）模拟分子和化学物质之间的相互作用。

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).