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

算符分裂或分裂步长方法(SSM)被广泛用于数值求解含时偏微分方程组，从流体力学到量子力学。为了使计算时间最小化，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设定的。Von Neumann分析被用来获得系数为常数的模型问题的上界。然而，实际有趣的方程的解在空间中通常不是恒定的。为了证明冯·诺依曼分析用于这类问题的合理性，人们通常用常系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，PI提出了另一种方法来分析SSM的不稳定性，当这种方法被用来模拟非线性薛定谔方程的孤子附近的解(即钟形解)时。在这个项目中，PI将把这种分析扩展到更实际的相关环境，涉及两个应用：光纤电信和玻色-爱因斯坦凝聚体。这将有助于理解本质上为变系数的问题中数值不稳定性的发展。然后，他们将使用这些信息来提出放宽稳定性要求的SSM的修改建议。显然，这将减少计算时间。这个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。为了准确地模拟感兴趣的物理过程，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量来逼近模拟过程。PI不会使用这种近似值，因为他们已经证明，这会导致对SSM性能的错误预测。他们的替代方法将依赖于数值分析和线性微分方程组理论的技术组合。它将帮助您了解SSM的性能限制。反过来，这将使他们能够对这种数值方法提出更有效和更可靠的修改。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体的建模。然而，他们的方法将影响SSM的其他应用，包括环境模拟、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)来模拟分子和化学物种之间的相互作用。

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, the PIs 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, the PIs 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. They 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. The PIs will not use this approximation, as they have demonstrated that it leads to incorrect predictions regarding the performance of the SSM. Their 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 them 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, their 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).