stable galerkin reduced order models for linearized compressible flow the galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations the numerical stability of galerkin models is shown to depend on the choice of inner product for the projection for the linearized euler equations a symmetry transformation leads to a stable formulation for the inner product boundary conditions for compressible flow that preserve stability of the reduced order model are constructed preservation of stability for the discrete implementation of the galerkin projection is made possible using a piecewise smooth finite element basis stability of the reduced order model using this approach is demonstrated on several model problems where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

the connection matrix theory for semiflows on not necessarily locally compact metric spaces abstract the index theory of rybakowski for isolated invariant sets and attractor repeller pairs in the setting of a semiflow on a not necessarily locally compact metric space is extended to include a connection matrix theory for morse decompositions partially ordered morse decompositions and attractor semifiltrations of invariant sets are defined and shown to be equivalent the definition and proof of existence of index filtrations for an ordered morse decomposition is provided via the index filtration the homology index braid and the connection matrices of the morse decomposition are defined

reduced order models based on local pod plus galerkin projection 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

an entire solution of a bistable parabolic equation on r with two colliding pulses we consider semilinear parabolic equations of the form equation a ut uxx f u x r t i where i 0 i 0 or i i solutions defined for all x t r2 x t r2 are referred to as entire solutions assuming that f c1 r f c1 r is of a bistable type with stable constant steady states 0 and 0 0 we show the existence of an entire solution u x t u x t of the following form for t t u t u t has two humps or pulses one near the other near as t increases the humps move toward the origin x 0x 0 eventually colliding and forming a one hump final shape of the solution with respect to the locally uniform convergence the solution u t u t is a heteroclinic orbit connecting the stable steady state 0 to the unstable ground state of the equation uxx f u 0uxx f u 0 we find the solution u as the limit of a sequence of threshold solutions of the cauchy problem for equation a

reduced order model for viscous aerodynamic flow past an airfoil a method is presented to obtain reduced order models of multiparametric steady transonic aerodynamic flows around two dimensional airfoils the method is based on proper orthogonal decomposition and appropriate projection on the proper orthogonal decomposition manifold of the governing equations and boundary conditions and it includes various ingredients that are new in this field proper orthogonal decomposition modes are obtained from a set of snapshots calculated by computational fluid dynamics based on the compressible navier stokes equations and a turbulence model but projection on the proper orthogonal decomposition manifold is made using the inviscid euler equations which makes the method independent of the computational fluid dynamics scheme projection is made using only a limited number of mesh points obtaining quite good results shock waves are treated specifically in the proper orthogonal decomposition description to avoid the need of using a too large number of snapshots the method is checked and discussed on a specific aerodynamic problem

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

on the use of pod based roms to analyze bifurcations in some dissipative systems abstract this paper deals with the use of pod based reduced order models to construct bifurcation diagrams which requires calculating steady and time dependent attractors in complex bifurcation problems involving dissipative systems the method proposed in the paper relies on the observation that pod manifolds resulting from snapshots calculated in time dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters the reason for this property is explained for a general class of dissipative systems which includes many problems of scientific industrial interest the consequence is that appropriate pod manifolds can be calculated in a quite computationally efficient way the method is illustrated considering both a simple bifurcation problem for a fisher like equation and a fairly complex bifurcation problem for the complex ginzburg landau equation

scalar parabolic pdes and braids the comparison principle for scalar second order parabolic pdes on functions u t x admits a topological interpretation pairs of solutions u 1 t 1 and u 2 t evolve so as to not increase the intersection number of their graphs we generalize to the case of multiple solutions u t n 1 by lifting the graphs to legendrian braids we give a global version of the comparison principle the curves u t evolve so as to weakly decrease the algebraic length of the braid we define a morse type theory on legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic pdes this is done via discretization to a finite dimensional system and a suitable conley index for discrete braids the result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic pdes we give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time periodic solutions

algorithms defined by nash iteration some implementations via multilevel collocation and smoothing abstract we describe the general algorithms of nash iteration in numerical analysis we make a particular choice of algorithm involving multilevel collocation and smoothing our test case is that of a linear differential equation although the theory allows for the approximate solution of nonlinear differential equations we describe the general situation completely and employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps we had earlier shown how these ideas apply to scattered data approximation but in this work we are interested in the application of the ideas to the numerical solution of differential equations we make use of approximate smoothers involving the solution of evolution equations with calibrated time steps

adaptive pod based low dimensional modeling supported by residual estimates summary 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

the connection map for attractor repeller pairs on montre que l application de connexion est additive les proprietes de l application connexion conduisent a une caracterisation des ensembles invariants isoles avec des points critiques hyperboliques comme une paire attracteur repulseur

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

a permutation characterization of sturm global attractors of hamiltonian type abstract we consider neumann boundary value problems of the form u t u x x f on the interval 0 x for dissipative nonlinearities f f u a permutation characterization for the global attractors of the semiflows generated by these equations is well known even in the much more general case f f x u u x we present a permutation characterization for the global attractors in the restrictive class of nonlinearities f f u in this class the stationary solutions of the parabolic equation satisfy the second order ode v f v 0 and we obtain the permutation characterization from a characterization of the set of 2 periodic orbits of this planar hamiltonian system our results are based on a diligent discussion of this mere pendulum equation

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

realization of meander permutations by boundary value problems abstract we consider neumann boundary value problems of the form uxx f x u ux 0 on the unit interval 0 x 1 for a certain class of dissipative nonlinearities f associated to these problems we have i meanders in the phase space u ux r 2 which are connected oriented simple curves on the plane intersecting a fixed oriented line the u axis in n points corresponding to the solutions and ii meander permutations f s n obtained by ordering the intersection points first along the u axis and then along the meander the meander permutation f is the permutation defined by the braid of solutions in the space x u ux it was recently shown by fiedler and rocha that f determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation ut uxx f x u ux up to c0 orbit equivalence therefore these permutations are of considerable importance in the classification problem of the morse smale attractors for these dynamical systems in this paper we present a purely combinatorial characterization of the set of meander permutations that are realizable by the above boundary value problems

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

spatially monotone homoclinic orbits in nonlinear parabolic equations of super fast diffusion type this work deals with positive classical solutions of the degenerate parabolic equation u t u p u xx quad quad star when p 2 which via the substitution v u1 p transforms into the super fast diffusion equation v t v m 1 v x x with m frac 1 p 1 in 1 0 it is shown that star possesses some entire positive classical solutions defined for all t in mathbb r and x in mathbb r which connect the trivial equilibrium to itself in the sense that u x t 0 both as t and as t locally uniformly with respect to x in mathbb r moreover these solutions have quite a simple structure in that they are monotone increasing in space the approach is based on the construction of two types of wave like solutions one of them being used for t 0 and the other one for 0 t both types exhibit wave speeds that vary with time and tend to zero as t and t respectively the solutions thereby obtained decay as x uniformly with respect to t in mathbb r but they are unbounded as x it is finally shown that within the class of functions having such a behavior as x there does not exist any bounded homoclinic orbit

adaptive non intrusive reduced order modeling for compressible flows abstract an adaptive non intrusive reduced basis rb method based on gaussian process regression gpr is proposed for parametrized compressible flows adaptivity is pursued in the offline stage the reduced basis by proper orthogonal decomposition pod is constructed iteratively to achieve a specified tolerance for gpr active data selection is used at each iteration with standard deviation as the error indicator to improve accuracy for shock dominated flows a properly designed simplified problem sp is considered as input of the regression models in addition to using parameters directly furthermore a surrogate error model is constructed to serve as an efficient error estimator for the gpr models several two and three dimensional cases are conducted including the inviscid nozzle flow the inviscid naca0012 airfoil flow and the inviscid m6 wing flow for all the cases the trained models are able to make efficient predictions with reasonable accuracy in the online stage the sp based approach is observed to result in biased sampling towards transonic regions the regression models are further applied in sensitivity analysis from which the solution of the two dimensional cases are shown to be significantly more sensitive to input parameters than the wing flow this is consistent to the comparison of convergence histories between the parameter based and the sp based models for cases of high sensitivity the sp based approach is superior and can help to significantly reduce the number of required snapshots to achieve a prescribed tolerance

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

calibrated reduced order pod galerkin system for fluid flow modelling to improve the behaviour of reduced order proper orthogonal decomposition pod galerkin systems two numerical methods are proposed these methods determine free parameters in the pod galerkin system from flow simulations via a minimization problem they give rise to linear systems and their computational costs are reasonable both methods are assessed for two flow configurations a two dimensional flow around a square cylinder for a reynolds number of 100 and a three dimensional flow past a backward facing step for a reynolds number of 7432 based on the step height and the streamwise velocity at the middle of the inlet for both configurations the methods are effective since accurate calibrated reduced order pod galerkin systems are obtained

equation free galerkin free pod assisted computation of incompressible flows we present a galerkin free proper orthogonal decomposition pod assisted computational methodology for numerical simulations of the long term dynamics of the incompressible navier stokes equations the approach is based on the equation free framework we use short appropriate initialized bursts of full direct numerical simulations dns of the navier stokes equations to observe estimate and accelerate through projective integration the evolution of the flow dynamics the main assumption is that the long term dynamics of the flow lie on a low dimensional attracting and invariant manifold which can be parametrized not necessarily spanned by a few pod basis functions we start with a discussion of the consistency and accuracy of the approach and then illustrate it through numerical examples two dimensional periodic and quasi periodic flows past a circular cylinder we demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long term asymptotic state of the flow in cases where traditional galerkin pod models fail the approach trades the overhead involved in developing pod galerkin and pod nonlinear galerkin codes for the repeated yet short and on demand use of an existing full dns simulator moreover since in this approach the pod modes are used to observe rather than span the true system dynamics the computation is much less sensitive than pod galerkin to values of the system parameters e g the reynolds number and the particular simulation data ensemble used to obtain the pod basis functions

a method to generate computationally efficient reduced order models a new method is presented to generate reduced order models roms in fluid dynamics problems the method is based on the expansion of the flow variables on a proper orthogonal decomposition pod basis calculated from a limited number of snapshots which are obtained via computational fluid dynamics cfd then the pod mode amplitudes are calculated as minimizers of a properly defined overall residual of the equations and boundary conditions the residual can be calculated using only a limited number of points in the flow field which can be scattered either all over the whole computational domain or over a smaller projection window this means that the process is both computationally efficient reconstructed flow fields require less than 1 of the time needed to compute a full cfd solution and flexible the projection window can avoid regions of large localized cfd errors also various definitions of the residual are briefly discussed along with the number and distribution of snapshots the number of retained modes and the effect of cfd errors to conclude that the method is numerically robust this is because the results are largely insensitive to the definition of the residual to cfd errors and to the cfd method itself which may contain artificial stabilizing terms thus the method is amenable for practical engineering applications