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).
