Topics in Computational Dynamics

计算动力学主题

• 批准号: 1419047
• 负责人: Erik Van Vleck
• 金额: $ 22.5万
• 依托单位: University of Kansas Center for Research Inc
• 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: 2014-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419047&HistoricalAwards=false
• 关键词: Topics; Computational; Dynamics

Models of complex systems often involve nonlinear differential equations that must be approximated numerically to gain detailed information. To understand the behavior of such approximations, this project explores similarities and differences in the dynamic behavior of the original model and what is obtained through numerical simulations. Understanding when the original model and numerical simulations yield similar dynamic behavior leads to greater confidence when making inferences from simulation results. In addition, stability analysis is useful in understanding the robustness of complex biological and physical phenomena. Of particular emphasis in this project is the application of these results to the dynamic behavior of models of climate dynamics. The techniques to be developed include approaches to dimension reduction and uncertainty quantification based upon Lyapunov exponent theory. The understanding of biological and physical processes will increase due to the computational techniques developed to address the dynamic behavior of detailed, microscopic models. In many areas of science and engineering dynamical systems are employed as models of complex phenomena. The focus of this project is on approximation of solutions of nonlinear dynamical systems. In particular, the investigator and colleagues are interested in understanding the dynamics of the finite dimensional approximations and how they relate to the dynamics of the original dynamical system. The main research topics to be investigated are related to the application of time dependent orthogonal change of variables and in the dynamics of traveling waves under discretization. Specific interests include the use of Lyapunov vectors (analogues of eigenvectors in time dependent stability analysis) for dimensional reduction of dissipative nonlinear differential equations, robust stability for time dependent linear Hamiltonian systems, and techniques for analyzing time dependent stability of numerical time stepping techniques, stability of plane wave solutions to bistable reaction-diffusion equations in periodic media, the impact of time and space discretization on traveling waves, and non-planar traveling waves in discrete media. Techniques to investigate these systems combine numerical analysis and dynamical systems ideas to gain a better understanding of computational dynamics both qualitatively and quantitatively.

复杂系统的模型通常涉及非线性微分方程，必须通过数值近似来获得详细信息。为了理解这种近似的行为，该项目探讨了原始模型的动态行为的相似性和差异，以及通过数值模拟获得的结果。了解原始模型和数值模拟何时产生相似的动态行为，可以在从模拟结果进行推断时提高置信度。此外，稳定性分析有助于理解复杂生物和物理现象的鲁棒性。在这个项目中特别强调的是这些结果的气候动力学模型的动态行为的应用。待开发的技术包括基于李雅普诺夫指数理论的降维和不确定性量化方法。生物和物理过程的理解将增加由于计算技术的发展，以解决详细的，微观模型的动态行为。在科学和工程的许多领域中，动力系统被用作复杂现象的模型。这个项目的重点是非线性动力系统的近似解。特别是，研究人员和同事们有兴趣了解有限维近似的动力学，以及它们如何与原始动力系统的动力学相关。主要的研究课题是与时间相关的正交变量变换的应用和离散化条件下的行波动力学。具体利益包括使用李雅普诺夫向量（在依赖于时间的稳定性分析中的特征向量的类似物），用于耗散非线性微分方程的降维，依赖于时间的线性Hamilton系统的鲁棒稳定性，以及用于分析数值时间步进技术的依赖于时间的稳定性的技术，周期介质中的反应扩散方程的平面波解的稳定性，时间和空间离散化对行波的影响，以及离散介质中的非平面行波。研究这些系统的技术结合了联合收割机数值分析和动力系统的思想，以更好地理解计算动力学的定性和定量。