Fully Online ROMs and Collocation Based on LUPOD

This paper deals with the acceleration of time-dependent solvers for nonlinear dissipative systems. The governing equations are Galerkin-projected onto a set of modes, which are obtained by applying proper orthogonal decomposition (POD) to a set of snapshots calculated by a standard numerical solver. The advantage of this approach is that the online operation of the resulting Galerkin system should be much faster than the standard numerical solver. The basic version of such reduced order model uses snapshots computed in a preprocess that is usually very computationally expensive. This difficulty can be overcome by an adaptive combination of the standard numerical solver and the Galerkin system along the simulation, using a method called POD on the Fly, which will be illustrated in a representative application. In addition, Galerkin projection can be performed using only a suitable set of collocation points, which decreases even further the computational cost. In this context, an efficient collocation method called LUPOD will be described and tested in various applications, including its combination with Galerkin projection.

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