Mathematical Sciences: Numerical Aspects of Riccati Transformation, Invariant Manifold Approximation, and Connected Issues

首席研究员将对一些 与黎卡提有关的分析和计算问题 变换，超稳定现象的整合， 刚性微分方程和不变流形 近似 这项工作包括：（一）进一步研究， 实现和软件生产的可靠算法， 积分微分Riccati方程，（ii）使用 微分Riccati方程进行刚性网格选择 两点边值问题的系统，（iii） 非线性两点边值问题程序软件实现 基于Riccati方法的值问题。 他将进一步 研究所谓的超稳定现象， 刚性微分方程的自适应积分，寻求 避免实际超稳定性的有效技术 发生的事件，并可以纳入一般用途 常微分方程求解器 最后，他会 继续研究不变量的数值近似 流形，实际上是对相关的数值解 双曲型偏微分方程组 这 涉及（i）非线性情况下的近似分析，（ii） 有效的解决方案技术（多重网格方法）， （三）利用与非线性系统的联系， Riccati变换 大多数真实的物理现象都是由 复杂的非线性微分方程组。 这些系统的精确解是无望的， 模拟是必要的。 这项工作的目的是生产可靠， 并严格证明，算法来解决其中的一些问题 系统. 其目的是生产软件工具， 工作科学家 特别是，研究黎卡提方程 将是有价值的人在游戏的理论工作，最优 控制和连接工程领域。 关于所谓的 刚性微分方程在求解系统时是有用的 具有不同的时间尺度，例如在化学动力学中， 空气动力学应用 最后，本文讨论了近似的工作。 不变流形有助于理解长期的 非线性动力系统的行为。 这有应用 在许多应用科学领域。

The principal investigator will conduct research on some analytical and computational problems related to the Riccati transformation, the superstability phenomenon for integration of stiff differential equations, and invariant manifold approximation. This work involves (i) further study, implementation and software production of reliable algorithms for integrating differential Riccati equations, (ii) using the differential Riccati equation to perform mesh-selection for stiff systems of two-point boundary value problems, (iii) software production of a code for nonlinear two-point boundary value problems based on the Riccati method. He will further study the so-called superstability phenomenon arising during adaptive integration of stiff differential equations, seeking efficient techniques that avoid the practical superstability occurrences and that can be incorporated into general purpose ordinary differential equation solvers. Finally, he will continue working on numerical approximation of invariant manifolds, in fact on the numerical solution of the associated systems of hyperbolic partial differential equations. This involves (i) approximation analysis for the nonlinear case, (ii) efficient solution techniques (multigrid approaches) for the resulting linear systems, (iii) exploitation of the link with the Riccati transformation. Most realistic physical phenomena are modelled by complicated nonlinear systems of differential equations. Exact solution of these systems is hopeless, and numerical simulation is a necessity. This work aims at producing reliable, and rigorously justified, algorithms to solve some of these systems. The intent is to produce software tools for the working scientist. In particular, work on the Riccati equation will be of value to people working in Games' Theory, Optimal Control and connected Engineering fields. The work on so-called stiff differential equations is useful in solving systems having different time-scales, such as in chemical kinetics and aerodynamics applications. Finally, the work on approximation of invariant manifolds is useful in understanding the long-term behavior of nonlinear dynamical systems. This has applications in many areas of applied science.