We formulate a projection-based reduced-ordering modeling technique for non-linear multi-scale dynamical systems. The proposed technique is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. The Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse scales. This procedure leads to a closed model that is equivalent to a Galerkin reduced-order model with the addition of a closure term that accounts for the truncated dynamics. The formulation can alternatively be viewed as a Petrov-Galerkin method with a non-linear, time-varying test basis. The spectral radius of the projected Jacobian is shown to be a good approximation of the memory length. Numerical experiments on the compressible Navier-Stokes equations in one and two-dimensions demonstrate that the proposed method leads to improvements over the standard Galerkin ROM and, in some cases, over the least-squares P 查看全文>>