Mathematical Sciences: Numerical Analysis for Time-Dependent Differential Equations

研究者斯图尔特研究了随时间变化的问题的数值逼近，这些问题是由普通或偏微发展方程模拟的。在这方面出现了两个基本问题。第一个问题是，给出一个随时间演变的问题的数值模拟，所获得的数据应该如何解释：在什么情况下，在什么意义上，计算的数据与基本微分方程产生的数据接近？该项目的第一个组成部分是在软件代码使用的复杂算法的背景下研究这个问题；这主要是一项分析性研究，需要了解由这些算法产生的不连续动力系统。第二个基本问题是如何有效地设计数值方法，使给定的关于方程的问题能够有效地得到回答，并产生新的见解。该项目的第二个组成部分是在耗散偏微分方程组的背景下研究这个问题，特别是发展轨道连接的计算技术；研究金兹堡-朗道方程和Kuramoto-Sivashinsky方程等方程。科学和工程中的许多问题都可以用演化微分方程来模拟。在实践中，这些方程不能精确求解，必须寻求计算近似。在这方面出现了两个重要问题：(1)“计算机生成的数据与我们希望了解的实际问题之间有什么关系”；(2)“如何开发有效的算法，以获得对感兴趣的问题最有意义的那类信息。”本项目在各种不同的背景下解决了这两个问题。首先，研究了一类非常广泛的算法，这些算法可能会对许多科学和工程领域产生影响，包括分子动力学模拟、行星相互作用模拟和其他有趣的进化问题。对于第二个问题，我们研究了一些特殊的问题，例如，流体向湍流转变的模型和复杂燃烧过程的模型。

Stuart The investigator studies the numerical approximation of problems that change in time and are modeled by ordinary or partial differential evolution equations. Two fundamental issues arise in this context. The first is, given a numerical simulation of a problem that evolves in time, how should the data obtained be interpreted: under what circumstances, and in what sense, is the computed data close to the data generated by the underlying differential equation? The first component of the project is concerned with studying this question in the context of the complicated algorithms used by software codes; this is primarily an analytical study requiring an understanding of the discontinuous dynamical systems generated by these algorithms. The second fundamental issue is how to effectively design numerical methods that enable given questions about the equations to be answered efficiently and in a way that yields new insights. The second component of the project is concerned with studying this question in the context of dissipative partial differential equations and, in particular, developing computational techniques for connecting orbits; equations such as the Ginzburg-Landau equation and Kuramoto-Sivashinsky equation are studied. Many problems in science and engineering can be edfectively modeled by means of differential equations of evolution. In practice these equations cannot be solved exactly and computational approximations must be sought. Two important issues arise in this context: (i) ``what is the relationship between computer generated data and the real problem we wish to understand'' and (ii) `` how can efficient algorithms be developed to obtain the type of information that sheds most light on the questions of interest.'' These two issues are addressed in this project in a variety of different contexts. For the first a very broad class of algorithms is studied with potential impact on many areas of science and engineering, including molecular dynamics simulations, planetary interaction simulations and other interesting evolution problems. For the second a number of particular problems are studied that arise from, for example, models of transition to turbulence in fluids and models of complicated combustion processes.