A local proper orthogonal decomposition (POD) plus Galerkin projection method is applied to the unsteady lid-driven cavity problem, namely the incompressible fluid flow in a two-dimensional box whose upper wall is moved back and forth at moderately large values of the Reynolds number. Such a method was recently developed for one-dimensional parabolic problems. Its extension to fluid dynamics problems is nontrivial (especially if rough CFD codes are used) and consists of using a computational fluid dynamics (CFD) code and a Galerkin system (GS) in a sequence of interspersed intervals $I_{CFD}$ and $I_{GS}$, respectively. The POD manifold is calculated retaining the most energetic POD modes resulting from the snapshots computed in the $I_{CFD}$ intervals; in fact, the POD manifold is completely calculated in the first $I_{CFD}$ interval but only updated in subsequent $I_{CFD}$ intervals. Intended to mimic industrial solvers, the CFD code contains unphysical terms that are introduced for purely numerical reasons to accelerate runs. The use of such a CFD code poses the question of which equations should be Galerkin projected onto the POD manifold, the exact governing equations or the CFD numerical scheme. Also, Galerkin projection can be made using either the standard $L_2$ inner product or a nonstandard one, based on a limited number of mesh points. After addressing these issues, a method is constructed that is able to accelerate the CFD code by a factor of the order of 5–15, depending on the Reynolds number and the nature (steady, periodic, or quasi-periodic) of the forcing velocity.

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.

An adaptive low-dimensional model is considered to simulate time-dependent dynamics in nonlinear dissipative systems governed by PDEs. The method combines an inexpensive POD-based Galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the POD modes. Switching between the numerical solver and the Galerkin system is decided ‘on the fly’ by monitoring (i) a truncation error estimate and (ii) a residual estimate. The latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental Galerkin system based on a larger number of POD modes. The most computationally expensive run of the numerical solver occurs at the outset, when the whole set of POD modes is calculated. This step is improved by using mode libraries, which may either be generic or result from former applications of the method. The outcome is a flexible, robust, computationally inexpensive procedure that adapts itself to the local dynamics by using the faster Galerkin system for the majority of the time and few, on demand, short runs of a numerical solver. The method is illustrated considering the complex Ginzburg–Landau equation in one and two space dimensions. Copyright © 2015 John Wiley & Sons, Ltd.

A recent conceptual model of tumor-driven angiogenesis including branching, elongation, and anastomosis of blood vessels captures some of the intrinsic multiscale structures of this complex system, yet allowing one to extract a deterministic integro-partial-differential description of the vessel tip density [ Phys. Rev. E 90, 062716 (2014)]. Here we solve the stochastic model, show that ensemble averages over many realizations correspond to the deterministic equations, and fit the anastomosis rate coefficient so that the total number of vessel tips evolves similarly in the deterministic and ensemble-averaged stochastic descriptions.

A data-driven modal analysis of plasma oscillations in a stationary plasma thruster-100-like Hall thruster in the 1–120 kHz range is presented. Data are generated by a two-dimensional (axial-radial) hybrid particle-in-cell/fluid simulation code. While proper orthogonal decomposition is unable to successfully uncouple the different dynamics, higher order dynamic mode decomposition (HODMD) cleanly isolates the breathing and ion transit time (ITT) modes. Indeed, the computed HODMD components can be clustered into two distinct groups, enabling the separate reconstruction of the dynamics of the two oscillation modes. It is also shown that each plasma variable exhibits a different behavior in each cluster. The breathing oscillations have a global or standing-wave character for the plasma density, neutral density and ion current density inside the thruster, and an axially progressive-wave structure for the electron temperature and plasma potential in the near plume. The ITT mode shows a progressive-wave structure for all variables, except for the neutral density whose amplitude is negligible. This work serves as an illustration of the applicability of data-driven techniques like HODMD to the analysis of plasma flows relevant to electric propulsion.