Approximation and Computation of Lyapunov Exponents, Global Errors, and Functional Traveling Waves

Van Vleck9973393研究员和他的同事们研究了计算微分方程中的问题：初值问题的阴影和相关的全局误差分析，稳定性信息和Lyapunov指数的计算，以及格型微分方程的分析和计算，即在空间上离散的和在时间上连续的微分方程。这个项目的中心主题是将动力系统思想与数值分析思想相结合。重点是开发和分析高效、准确的数值技术，以及有助于识别微分方程有趣的动态特性的分析和计算技术。研究人员和他的同事研究了解在微分方程的数值计算过程中发生的误差传播的方法。这种理解是基本的，因为物理和生物系统的模型通常是使用计算技术来求解的。格点微分方程组的研究在材料模拟领域有着广泛的应用。正在考虑的模型允许包括微观和宏观效应，随着计算机技术的进步允许更大、更复杂的材料模型，这将变得更加重要。

Van Vleck9973393The investigator and his colleagues study issues in computational differential equations: shadowing and related topics of global error analysis for initial value problems, computation of stability information and Lyapunov exponents, and analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time. This project has as a central theme the combination of dynamical systems ideas with numerical analysis ideas. The focus is on the development and analysis of efficient, accurate numerical techniques and on analytical and computational techniques that help identify interesting dynamic properties of differential equations.The investigator and his colleagues study methods for understanding the propagation of errors that occur during the numerical computation of differential equations. This understanding is fundamental because models of physical and biological systems are often solved using computational techniques. The research on lattice differential equations has applications in the area of materials modeling. The models being considered allow for the inclusion of both microscopic and macroscopic effects, which will become more important as advances in computer technology allow for larger, more sophisticated models of materials.