Jolly The investigator and his colleague test approximate inertial manifolds (AIMs) on nonlinear dissipative evolution equations, including the Kuramoto-Sivashinsky, Navier-Stokes, and Lorenz equations. The emphasis is on capturing elements of the global attractor, and not on solving the initial value problem over short time intervals. The phase space dimension reduction offered by AIMs is exploited to visualize global bifurcations involving two-dimensional stable manifolds, which would be of higher dimension in traditional Galerkin discretizations. Of particular interest is an algebraic AIM, which offers arbitrary accuracy, at a fixed dimension. Increased accuracy in this case is achieved by increasing the degree of a polynomial that implicitly defines the AIM. The investigators explore various time discretizations that take advantage of the particular features of the associated differential-algebraic system. This research has an impact on how the long term behavior of physical systems is determined by computers. Part of the work is devoted to comparing the efficiency of alternative algorithms to standard ones used in high performance computing. Another is to visualize certain critical phenomena, which could not be seen with traditional methods, regardless of the computational effort. While the various methods will be tested on particular mathematical models of combustion, fluid flow, and the weather, they are applicable to a wide range of problems that fit into a general framework.
