Methods for dimension reduction and their application in nonlinear dynamics

Abstract

We compare linear and nonlinear Galerkin methods in their efficiency to reduce infinite dimensional systems, described by partial differential equations, to low dimensional systems of ordinary differential equations, both concerning the effort in their application and the accuracy of the resulting reduced system. Important questions like the choice of the form of the ansatz functions (modes), the choice of the number m of modes and, finally, the construction of the reduced system are addressed. For the latter point, both the linear or standard Galerkin method making use of the Karhunen Loeve (proper orthogonal decomposition) ansatz functions and the nonlinear Galerkin method, using approximate inertial manifold theory, are used. In addition, also the post-processing Galerkin method is compared with the other approaches.