- March 2014
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- An adaptive method is presented for the construction of bifurcation diagrams providing the large time dynamics of dissipative systems as a bifurcation parameter is varied. The method combines a standard numerical solver and a Galerkin system resulting from Galerkin projecting the governing equations onto a set of proper orthogonal decomposition (POD) modes that are computed on the fly as the bifurcation parameter is varied. The numerical solver provides some snapshots that are used to either completely calculate (at the outset of the process) or update (later on) the POD modes. This solver is run over short time intervals only, at specific values of the bifurcation parameter, meaning that the (much faster) Galerkin system is used for the majority of the calculation. The key ingredients are (i) detecting on the fly when updating is needed and (ii) reconstructing the POD modes. The necessity of updating the POD basis is detected monitoring (a) the amplitudes of the high-order modes (which estimate the accuracy of the approximation) and (b) consistency with the solution provided by a second instrumental Galerkin system that retains a larger number of POD modes (which detects truncation instabilities of the first Galerkin system). Updating the POD modes is performed applying POD to a mix of appropriately weighted old and new modes. The method is tested on the complex Ginzburg-Landau equation, considering bifurcation diagrams that exhibit a variety of bifurcations involving periodic, quasi-periodic, and chaotic attractors, and turns out to be flexible, robust, and computationally fast.
- reduced-order models; adaptive methods; galerkin projection; proper orthogonal decomposition; poincare maps; bifurcation diagrams; complex ginzburg-landau equation; reduced-order-model; inertial manifolds; fluid-flow; computation; dynamics; reduction; equation; systems; generate; symmetry