This paper deals with an extension of dynamic mode decomposition (DMD), which is appropriate to treat general periodic and quasi-periodic dynamics, and transients decaying to periodic and quasi-per...

A method is presented to accelerate numerical simulations on parabolic problems using a numerical code and a Galerkin system (obtained via POD plus Galerkin projection) on a sequence of interspersed intervals. The lengths of these intervals are chosen according to several basic ideas that include an a priori estimate of the error of the Galerkin approximation. Several improvements are introduced that reduce computational complexity and deal with: (a) updating the POD manifold (instead of calculating it) at the end of each Galerkin interval; (b) using only a limited number of mesh points to calculate the right hand side of the Galerkin system; and (c) introducing a second error estimate based on a second Galerkin system to account for situations in which qualitative changes in the dynamics occur during the application of the Galerkin system. The resulting method, called local POD plus Galerkin projection method, turns out to be both robust and efficient. For illustration, we consider a time-dependent Fisher-like equation and a complex Ginzburg–Landau equation

This article shows the capability of using a higher order dynamic mode decomposition (HODMD) algorithm both to identify flow patterns and to extrapolate a transient solution to the attractor region. Numerical simulations are carried out for the three-dimensional flow around a circular cylinder, and both standard dynamic mode decomposition (DMD) and higher order DMD are applied to the non-converged solution. The good performance of HODMD is proved, showing that this method guesses the converged flow patterns from numerical simulations in the transitional region. The solution obtained can be extrapolated to the attractor region. This fact sheds light on the capability of finding real flow patterns in complex flows and, simultaneously, reducing the computational cost of the numerical simulations or the required quantity of data collected in experiments.

This paper presents a systematic numerical study of the effects of heat transfer and pressure drop produced by vortex promoters of various shapes in a 2D, laminar flow in a micro-channel. The liquid is assumed to be water, with temperature dependent viscosity and thermal conductivity. It is intended to obtain useful design criteria of micro-cooling systems, taking into account that practical solutions should be both thermally efficient and not expensive in terms of the pumping power. Three reference cross sections, namely circular/elliptical, rectangular, and triangular, at various aspect ratios are considered. The effect of the blockage ratio, the Reynolds number, and the relative position and orientation of the obstacle are also studied. Some design guidelines based on two figures of merit (related to thermal efficiency and pressure drop, respectively), which could be used in an engineering environment are provided

This paper deals with singular semilinear elliptic equations in bounded domains with Dirichlet boundary data. The elliptic operator is a second-order operator not necessarily in divergence form. We consider existence, uniqueness and linearized stability of positive solutions for a series of nonlinear eigenvalue problems.