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

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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.