Mathematical Sciences: The Numerical Analysis of Evolution Equations Over Long Time Intervals

研究人员将分析和设计数值方法 对于需要长时间求解的初值问题， 时间间隔。 将研究两类方程： 拥有一个吸收集，其中长时间动态可能是 非常复杂（混乱），但仍保持在一个有界的集合; 这是一个梯度结构，所以解决方案接近的集合， （通常是多个）稳态。 固定 时间步进策略，建议的研究领域是一个 A、B和G稳定性的经典理论的推广。 这些经典理论适用于具有简单 动力学和它是感兴趣的，以确定如何理论可以 可用于和修改，以科普更复杂的动态 问题 对于可变时间步进策略，一些现有的 线性问题的工作将推广到非线性问题 问题工作还将致力于 同宿轨道和异宿轨道，它们起着重要作用 在两类方程的长时间动力学中， 考虑. 物理科学中许多有趣的现象， 工程学要求理解动力学现象， 在很长的时间尺度上进化。 例子包括湍流 在流体中，在固体中的相分离，和行星运动。 这些现象通常由微分方程来模拟， 有必要通过计算近似这些方程， 获取潜在问题信息的技术 被模仿。 一个至关重要的问题是 确定计算机生成的输出的有效性， 特别是，重要的是要充分了解关系 在计算机生成的模型近似值之间， 模型本身的行为。 对这个问题的研究是 在许多简单的情况下都很容易理解，但这些情况并不 包括存在较长时间尺度情况。 的 本研究的目的是研究这一问题， 微分方程模型的基本理论指导 研究计算机生成的近似值。

The investigator will analyse and design numerical methods for initial value problems whose solutions are required over long time intervals. Two classes of equations will be studied: those possessing an absorbing set, where the long-time dynamics may be extremely complex (chaotic) but remain in a bounded set; and with those a gradient structure, so that solutions approach the set of (typically multiple) steady states for large time. For fixed time-stepping strategies, the proposed area of research is a generalisation of the classical theories of A, B and G stability. These classical theories apply to classes of problems with simple dynamics and it is of interest to determine how the theories can be used and modified to cope with more complicated dynamical problems. For variable time-stepping strategies some existing work on linear problems will be generalised to nonlinear problems. Work will also be directed towards the approximation of homoclinic and heteroclinic orbits, which play an important role in the long-time dynamics of the two classes of equation under consideration. Many phenomena of interest in the physical sciences and engineering require the understanding of dynamical phenomena that evolve over very long time scales. Examples include turbulence in fluids, phase separation in solids, and planetary motions. Often these phenomena are modelled by differential equations and it is necessary to approximate these equations by computational techniques to obtain information about the underlying problem being modelled. A question of fundamental importance is to ascertain the validity of the computer-generated output and, in particular, it is important to understand fully the relationship between the computer-generated approximations of the model and the behavior of the model itself. The study of this problem is well understood in many simple situations but these do not include situations where long time scales are present. The object of the research is to study this problem, using the underlying theory of the differential equation models to guide the study of the computer-generated approximations.