Computational and Analytical Challenges in Nonlinear Dispersive Wave Equations

非线性色散波动方程的计算和分析挑战

• 批准号: 1409018
• 负责人: Gideon Simpson
• 金额: $ 14.61万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1409018&HistoricalAwards=false
• 关键词: Computational; Analytical; Challenges; Nonlinear; Dispersive

Nonlinear dispersive wave equations model a variety of phenomena in fluid mechanics, plasma physics, turbulence, and optics. Analyzing and simulating such equations numerically allows us to make predictions pertaining to the corresponding application. For example, it would be desirable to address the following questions arising in applications: Will signals in optical devices, used for the transmission and storage of data, be stable, or how is energy transferred in turbulent systems? This project will advance our knowledge in two ways. First, it will seek to prove, in a mathematically rigorous sense, properties of the equations, an example of which is the stability of optical pulses. Second, it will develop and justify numerical algorithms for simulating the equations on computers. Systematic analysis of the algorithms ensures that the computer simulations truly reflect the equations. Finally, both the analysis and simulation of the equations will determine the regime of validity of these equations as models for the physical applications. A supplemental benefit of this research is that the numerical algorithms it develops will be applicable to a broad class of equations and physical problems. The equations under investigation in this project are of the nonlinear Schrödinger type, including the derivative nonlinear Schrödinger equation and the Gross-Pitaevskii equation. Major analytical questions this work will address include the existence and stability of singular solutions and nonlinear bound states. These problems will be addressed through asymptotic analysis, variational methods, and spectral theory. Singularity formation will also be investigated through simulation, using adaptive meshing methodology, such as iterative grid redistribution. Numerical algorithms for simulating the time dependent problems will be implemented by operator splitting methods, treating the nonlinear and linear dispersive parts of the equation separately. This numerical analysis will be pursued with the intention of constructing efficient energy preserving methods suitable for long time simulation. An algorithm for computing nonlinear bound states will be studied in a semi-discrete form, so as to be insensitive to spatial discretization. The bound state algorithms will assist in the construction of initial conditions, which can then be studied using the time dependent algorithms.

非线性色散波动方程模拟了流体力学、等离子体物理、湍流和光学中的各种现象。分析和模拟这样的方程数值使我们能够作出预测有关的相应应用。例如，希望解决应用中出现的以下问题：用于传输和存储数据的光学设备中的信号是否稳定，或者能量如何在湍流系统中传输？ 这个项目将从两个方面增进我们的知识。 首先，它将试图证明，在严格的数学意义上，方程的性质，其中的一个例子是光脉冲的稳定性。 其次，它将开发和证明在计算机上模拟方程的数值算法。 对算法的系统分析确保计算机模拟真实地反映方程。 最后，方程的分析和模拟将确定这些方程作为物理应用模型的有效性。 这项研究的一个补充好处是，它开发的数值算法将适用于广泛的一类方程和物理问题。 本项目研究的方程是非线性薛定谔方程，包括导数非线性薛定谔方程和Gross-Pitaevskii方程。 主要的分析问题，这项工作将解决包括奇异解和非线性束缚态的存在性和稳定性。 这些问题将通过渐近分析，变分方法和谱理论来解决。 奇异性的形成也将通过模拟研究，使用自适应网格划分方法，如迭代网格重新分配。 模拟时间相关问题的数值算法将通过算子分裂方法实现，分别处理方程的非线性和线性色散部分。 这种数值分析将追求的意图，构建有效的能量保存方法，适合长时间的模拟。 计算非线性束缚态的算法将以半离散形式进行研究，以便对空间离散不敏感。 束缚态算法将有助于初始条件的构建，然后可以使用时间相关算法进行研究。