University of Leicester

The concept of the slow invariant manifold is the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the constructive methods of invariant manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space. The equation of motion for immersed manifolds is obtained.

Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.

A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.

The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for nudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; model reduction in chemical kinetics.

Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients

possessing unique solutions make up a very important class in applications. For

instance, Langevin-type equations and gradient systems with noise belong to this

class. At the same time, most numerical methods for SDEs are derived under the

global Lipschitz condition. If this condition is violated, the behaviour of many

standard numerical methods in the whole space can lead to incorrect conclusions.

This situation is very alarming since we are forced to refuse many effective

methods and/or to resort to some comparatively complicated numerical procedures.

We propose a new concept which allows us to apply any numerical method of weak

approximation to a very broad class of SDEs with nonglobally Lipschitz

coefficients. Following this concept, we discard the approximate trajectories

which leave a sufficiently large sphere. We prove that accuracy of any method of

weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be

made arbitrarily small with increasing the radius of the sphere. The results

obtained are supported by numerical experiments. The concept of rejecting

exploding trajectories is applied to computing averages with respect to the

invariant law for Langevin-type equations. This approach to computing ergodic

limits does not require from numerical methods to be ergodic and even convergent

in the nonglobal Lipschitz case. The talk is based on joint papers with G.N.

Milstein.