GALERKIN PROJECTIONS AND THE PROPER ORTHOGONAL DECOMPOSITION FOR EQUIVARIANT EQUATIONS

In this paper we are interested in Galerkin projections of certain partial differential evolution equations, that retain the symmetries of the original nontruncated equations. In particular, we extend several results concerning the Galerkin projections, based on the proper orthogonal decomposition (which is also known as the Karhunen-Loeve expansion) for equations which are equivariant under the action of compact Abelian symmetry groups. We provide sufficient conditions for numerical schemes, based on the linear and the nonlinear Galerkin method, to retain the symmetries of the original system. We perform a rational analysis of methods which use symmetry to enlarge the ensemble size of a data set. Our analysis has the potential for computational savings.