Accurate Low-Dimensional Approximation of the Linear Dynamics of Fluid Flow

Abstract

Methods for approximating a stable linear autonomous dynamical system by a system of lower order are examined. Reducing the order of a dynamical system is useful theoretically in identifying the irreducible dimension of the dynamics and in isolating the dominant spatial structures supporting the dynamics, and practically in providing tractable lower-dimension statistical models for climate studies and error covariance models for forecast analysis and initialization. Optimal solution of the model order reduction problem requires simultaneous representation of both the growing structures in the system and the structures into which these evolve. For autonomous operators associated with fluid flows a nearly optimal solution of the model order reduction problem with prescribed error bounds is obtained by truncating the dynamics in its Hankel operator representation. Simple model examples including a reduced-order model of Couette flow are used to illustrate the theory. Practical methods for obtaining approximations to the optimal order reduction problem based on finite-time singular vector analysis of the propagator are discussed and the accuracy of the resulting reduced models evaluated.