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

主要研究人员将研究与Riccati变换、刚性微分方程积分的超稳定性现象和不变流形逼近有关的一些分析和计算问题。这项工作包括(I)进一步研究、实现和制作可靠的Riccati方程积分算法，(Ii)使用Riccati方程对刚性两点边值问题组进行网格选择，(Iii)基于Riccati方法的非线性两点边值问题程序的软件制作。他将进一步研究刚性微分方程自适应积分过程中出现的所谓超稳定性现象，寻找避免实际超稳定性发生并可并入通用常微分方程组解算器的有效技术。最后，他将继续研究不变流形的数值逼近，实际上是双曲型偏微分方程组的数值解。这涉及(I)非线性情况的近似分析，(Ii)所得到的线性系统的有效解技术(多重网格法)，(Iii)利用Riccati变换的联系。大多数现实的物理现象都是由复杂的非线性微分方程组来模拟的。这些系统的精确解是无望的，而数值模拟是必要的。这项工作的目的是产生可靠的、严格合理的算法来解决其中的一些系统。其目的是为工作的科学家生产软件工具。尤其是，Riccati方程的工作对博弈论、最优控制和相关工程领域的工作人员来说将是有价值的。所谓的刚性微分方程组的工作对于求解具有不同时间尺度的系统是有用的，例如在化学动力学和空气动力学应用中。最后，关于不变流形逼近的工作对于理解非线性动力系统的长期行为是有用的。这在应用科学的许多领域都有应用。

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.