Fast time integration of PDEs combining POD and Galerkin projection based on a limited set of mesh points

Resumen

Recently, an adaptive method to accelerate time dependent numerical solvers of systems of PDEs that require a high cost in computational time and memory has been proposed [3] (see also [1, 2]). The method combines on the fly such numerical solver with a proper orthogonal decomposition, from which we identify modes, a Galerkin projection (that provides a reduced system of equations), and the integration of the reduced system. The strategy is based on a truncation error estimate and a residual estimate, designed to control the truncation error and the mode truncation instability, respectively. These estimates support the selection of the appropiate time intervals in which the numerical solver is run to first construct and then update, on demand, the POD modes. Moreover, to reduce the computational effort needed at the outset to generate the initial POD subspace, information from former simulations or generic libraries (e.g. trigonometric functions or orthogonal polynomials) were also used. To improve the computational efficiency of the method presented in [3] a crucial step is to use a limited number of points (instead of the whole computational mesh used in the spatial discretization) to both perform POD and to Galerkin?project the equations. In this work we will discuss and compare several alternatives in representative examples illustrating that a suitable point selection can make the cost of the reduced order model (associated with POD, Galerkin projection and the integration of the resulting Galerkin system) negligible compared to that of the reference numerical solver.