stability radius and internal versus external stability in banach spaces an evolution semigroup approach in this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems these include autonomous and nonautonomous systems modeled with unbounded state space operators acting on banach spaces this approach allows one to apply the classical theory of strongly continuous semigroups to time varying systems in particular the complex stability radius may be expressed explicitly in terms of the generator of an evolution semigroup examples are given to show that classical formulas for the stability radius of an autonomous hilbert space system fail in more general settings upper and lower bounds on the stability radius are proven for banach space systems in addition it is shown that the theory of evolution semigroups allows for a straightforward operator theoretic analysis of internal stability as determined by classical frequency domain and input output operators even for nonautonomous banach space systems in particular for the nonautonomous setting internal stability is shown to be equivalent to input output stability for stabilizable and detectable systems for the autonomous setting an explicit formula for the norm of input output operator is given

isogeometric divergence conforming variational multiscale formulation of incompressible turbulent flows a new residual based variational multiscale rbvms formulation for incompressible turbulent flows is proposed that is suitable for discretization using divergence conforming b splines the proposed methodology results in a pointwise satisfaction of the zero divergence constraint on the discrete velocity field the velocity fine scales are residual driven and constructed in a manner that is consistent with the divergence free constraint on the discrete velocity solution the resulting formulation is tested on laminar and turbulent flow benchmark problems showing excellent stability and accuracy characteristics in both regimes

decay estimates of functions through singular extensions of vector valued laplace transforms abstract let x be a banach space and let f l r x whose laplace transform extends analytically to some region containing i r 0 possibly having a pole at the origin in this paper we give estimates of the decay of certain slight suitable modification of f in terms of the growth of its laplace transform along the imaginary axis this technique is applied to obtain decay estimates of smooth orbits of bounded c 0 semigroups whose infinitesimal generators have an arbitrary finite boundary spectrum these results are close to those given recently by c j k batty and t duyckaerts

stability theory for semigroups using l p l q fourier multipliers abstract we study polynomial and exponential stability for c 0 semigroups using the recently developed theory of operator valued l p l q fourier multipliers we characterize polynomial decay of orbits of a c 0 semigroup in terms of the l p l q fourier multiplier properties of its resolvent using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space we do not assume that the semigroup is uniformly bounded our results depend only on spectral properties of the generator as a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability and we also obtain a new theorem on exponential stability for positive semigroups

on coupled modes theory of two dimensional wave motion in elastic waveguides with slowly varying parameters in curvilinear orthogonal coordinates coupled modes theory as applied to 2 d wave propagation in range dependent isotropic elastic waveguides in curvilinear orthogonal coordinates is proposed the model consists of one or more curvilinear layers with interfaces and outer boundaries departing slowly from the coordinate surfaces the material parameters are supposed to be slowly dependent on the longitudinal coordinate four dimensional vector valued functions with the displacements rotation and divergence as the components are shown to be a convenient form of wave field description to extend the coupled modes theory from liquid to elastic waveguides for this case the cross section operator is defined and the generalized bi orthogonality property of its eigenfunctions is established using the generalized bi orthogonality the coupling equations are derived wave propagation in a tube which narrows slowly to a conic rod and in a disk of radially dependent thickness are considered as the examples the differences between the results obtained by the coupled modes theory and those the multiple scales theory leads to are discussed the theory modification for dealing with a cutoff phenomenon and the general algorithm of computation of the wave field excited by a point source are considered

coupled mode theory of intrinsic modes in a wedge recent theoretical studies of acoustic wave propagation in a model waveguide consisting of a homogeneous wedge with one reflecting and one penetrable boundary have established the utility of the concept of an intrinsic mode which generalizes for nonseparable problems the normal mode of separable configurations in this paper it is shown by explicit calculations on the same model problem that direct orthogonal expansion in local normal modes of the intrinsic mode produces the same expansion coefficients as those obtained by a perturbation analysis of the coupled mode equations the perturbation method used is that of renormalization in which the mode coupling operator is iteratively diagonalized up to a certain order in the nonseparability parameter which in this case is the wedge angle

the stability number of the timoshenko system with second sound abstract in this work we consider the timoshenko beam model with second sound we introduce a new number 0 that characterizes the exponential decay we prove that the corresponding semigroup associated to the system is exponentially stable if and only if 0 0 otherwise there is a lack of exponential stability in this case we prove that the semigroup decays as t 1 2 moreover we show that the rate is optimal

a hamiltonian method for finding broadband modal eigenvalues for shallow water waveguides over a layered elastic bottom modal eigenvalues can be determined by searching the locations in the complex plane of the horizontal wave number at which the complex phase function is a multiple of c t tindle and n r chapman j acoust soc am 96 1777 1782 1994 in this paper a hamiltonian method is introduced for tracing the path in the complex plane along which the phase function keeps real the hamiltonian method can also be extended to compute the broadband modal eigenvalues or the modal dispersion curves in the pekeris waveguide with fluid elastic bottoms for each proper or leaky normal mode a different hamiltonian is constructed in the complex plane and used to trace automatically the complex dispersion curve with the eigenvalue in a reference frequency as the initial value in contrast to the usual methods the dispersion curve for each mode is determined individually the hamiltonian method shows good performance by comparing with kraken

preconditioned nullspace method for the two dimensional oseen problem the oseen problem which arises in the simulation of the time dependent navier stokes equations for incompressible fluid flow leads to indefinite nonsymmetric and possibly ill conditioned linear systems of equations this paper presents a method for obtaining a reduced linear system from the original system which is then solved by the preconditioned bicgstab method the system reduction is obtained through an efficient implicit representation of a basis of discretely divergence free functions also known as the nullspace method we will show a close relationship between this preconditioned nullspace method and the bfbt preconditioned pressure schur complement problem arising in typical block preconditioners we will use this relationship to analyze the spectrum of the preconditioned reduced system we will also present numerical tests for the two dimensional oseen problem to illustrate the performance of this method which is similar to the performance of the bfbt preconditioned pressure schur complement solver in particular it is robust w r t the strength of the viscosity for simple constant wind but not for more complicated flows and it shows only a slight dependence on the mesh size

an efficient iterative method for the generalized stokes problem the generalized stokes problem which arises frequently in the simulation of time dependent navier stokes equations for incompressible fluid flow gives rise to symmetric linear systems of equations these systems are indefinite due to a set of linear constraints on the velocity causing difficulty for most preconditioners and iterative methods this paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method this new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite numerical experiments indicating superior convergence compared to existing methods are presented a natural extension of this method to elliptic problems is also proposed along with theoretical bounds on the rate of convergence and results of experiments demonstrating robust and effective preconditioning

boundary conditions for a divergence free velocity pressure formulation of the navier stokes equations new sets of boundary conditions for the velocity pressure formulation of the incompressible navier stokes equations are derived the boundary conditions have the same form on both inflow and outflow boundaries and lead to a divergence free solution moreover the specific form of the boundary condition makes it possible derive a symmetric positive definite equation system for the internal pressure numerical experiments support the theoretical conclusions

asymptotic behaviour of c0 semigroups and γ boundedness of the resolvent abstract let x be a banach space with type p and cotype q and let a be the infinitesimal generator of a c 0 semigroup on x we show that if the resolvent of a has a bounded analytic extension to c then a has negative fractional growth bound of order 1 p 1 q

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

a robust and accurate outflow boundary condition for incompressible flow simulations on severely truncated unbounded domains we present a robust and accurate outflow boundary condition and an associated numerical algorithm for incompressible flow simulations on unbounded physical domains aiming at maximizing the domain truncation without adversely affecting the flow physics the proposed outflow boundary condition allows for the influx of kinetic energy into the domain through the outflow boundaries and prevents un controlled growth in the energy of the domain in such situations the numerical algorithm for the outflow boundary condition is developed on top of a rotational velocity correction type strategy to de couple the pressure and velocity computations and a special construction in the algorithmic formulation prevents the numerical locking at the outflow boundaries for time dependent problems extensive numerical tests for flow problems with bounded and semi bounded physical domains demonstrate that this outflow boundary condition and the numerical algorithm produce stable and accurate simulations on even severely truncated computational domains where strong vortices may be present at or exit the outflow boundaries the method developed herein has the potential to significantly expedite simulations of incompressible flows involving outflow or open boundaries and to enable such simulations at reynolds numbers significantly higher than the state of the art

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

optimal rates of decay for operator semigroups on hilbert spaces abstract we investigate rates of decay for c 0 semigroups on hilbert spaces under assumptions on the resolvent growth of the semigroup generator our main results show that one obtains the best possible estimate on the rate of decay that is to say an upper bound which is also known to be a lower bound under a comparatively mild assumption on the growth behaviour this extends several statements obtained by batty et al 2016 6 in fact for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly we illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary

an improved finite element space for discontinuous pressures abstract we consider incompressible stokes flow with an internal interface at which the pressure is discontinuous as happens for example in problems involving surface tension we assume that the mesh does not follow the interface which makes classical interpolation spaces to yield suboptimal convergence rates typically the interpolation error in the l 2 norm is of order h 1 2 we propose a modification of the p 1 conforming space that accommodates discontinuities at the interface without introducing additional degrees of freedom or modifying the sparsity pattern of the linear system the unknowns are the pressure values at the vertices of the mesh and the basis functions are computed locally at each element so that the implementation of the proposed space into existing codes is straightforward with this modification numerical tests show that the interpolation order improves to o h 3 2 the new pressure space is implemented for the stable p 1 p 1 mini element discretization and for the stabilized equal order p 1 p 1 discretization assessment is carried out for poiseuille flow with a forcing surface and for a static bubble in all cases the proposed pressure space leads to improved convergence orders and to more accurate results than the standard p 1 space in addition two navier stokes simulations with moving interfaces rayleigh taylor instability and merging bubbles are reported to show that the proposed space is robust enough to carry out realistic simulations

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

error bounded schemes for time dependent hyperbolic problems in this paper we address the error growth in time for hyperbolic problems in first order form in bounded domains the energy method is used to study when an error growth or a fixed error bound is obtained it is shown that the choice of boundary procedure is a crucial point numerical experiments corroborate the theoretical findings

asymptotics for infinite systems of differential equations this paper investigates the asymptotic behavior of solutions to certain infinite systems of ordinary differential equations in particular we use results from ergodic theory and the asymptotic theory of c 0 semigroups to obtain a characterization in terms of convergence of certain cesaro averages of those initial values which lead to convergent solutions moreover we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition these results rely on a detailed spectral analysis of the operator describing the system which is made possible by certain structural assumptions on the operator the resulting class of systems is sufficiently broad to cover a number of important applications including in particular both the so called robot rendezvous problem and an important class of platoon systems arising in control theory our method leads to new results in both cases

a numerical method for ocean acoustic normal modes the method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans the propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field such errors can severely degrade the accuracy of the normal mode representation particularly at long ranges in this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes it consists of a combination of well known numerical procedures such as sturm sequences the bisection method newton s and brent s methods richardson extrapolation and inverse iteration we also introduce a modified richardson extrapolation procedure that substantially increases the speed and accuracy of the computation

rational operators for filtering the parabolic equation pe method has been applied to approximate the wave equation construct radiation conditions solve scattering problems construct initial conditions and derive energy flux conditions a new application of the pe method filters based on rational approximations of depth operators is described and tested the rational approximation is designed to act as the identity operator on the desired part of the spectrum of the depth separated wave equation and to annihilate other parts of the spectrum this approach does not require explicit knowledge of the spectrum the applications of the filter include directly solving eigenvalue problems annihilating components of the wave number spectrum generating initial conditions at the source range and eliminating gibbs oscillations that arise in energy conserving pe solutions

an integrability condition on the resolvent for hyperbolicity of the semigroup abstract in this paper we present necessary and sufficient conditions on the resolvent of the generator of a strongly continuous semigroup in order that the corresponding semigroup is hyperbolic the proof is based on a new characterization of the growth bound in terms of the resolvent as an application we discuss three examples

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

summation by parts operators for correction procedure via reconstruction the correction procedure via reconstruction cpr formerly known as flux reconstruction is a framework of high order methods for conservation laws unifying some discontinuous galerkin spectral difference and spectral volume methods linearly stable schemes were presented by vincent et al 2011 2015 but proofs of non linear entropy stability in this framework have not been published yet to the knowledge of the authors we reformulate cpr methods using summation by parts sbp operators with simultaneous approximation terms sats a framework popular for finite difference methods extending the results obtained by gassner 2013 for a special discontinuous galerkin spectral element method this reformulation leads to proofs of conservation and stability in discrete norms associated with the method recovering the linearly stable cpr schemes of vincent et al 2011 2015 additionally extending the skew symmetric formulation of conservation laws by additional correction terms entropy stability for burgers equation is proved for general sbp cpr methods not including boundary nodes

integral conditions for the pressure in the computation of incompressible viscous flows abstract the problem of finding the correct conditions for the pressure in the time discretized navier stokes equations when the incompressibility constraint is replaced by a poisson equation for the pressure is critically examined it is shown that the pressure conditions required in a nonfractional step scheme to formulate the problem as a system of split second order equations are of an integral character and similar to the previously discovered integral conditions for the vorticity the novel integral conditions for the pressure are used to derive a finite element method which is very similar to that developed by glowinski and pironneau and is the finite element counterpart of the influence matrix method of kleiser and schumann

source localization in a waveguide using polynomial rooting source localization in acoustic waveguides involves a multidimensional search procedure we propose a new algorithm in which the search in the depth direction is replaced by polynomial rooting using the proposed algorithm range and depth estimation by a vertical array requires a 1 d search procedure for a 3 d localization problem i e range depth and direction of arrival doa estimation the algorithm involves a 2 d search procedure consequently the proposed algorithm requires significantly less computation than other methods that are based on a brute force search procedure over the source location parameters in order to evaluate the performance of the algorithm an error analysis is carried out and monte carlo simulations are performed the results are compared with the cramer rao bound crb and to the maximum likelihood ml simulation performance the algorithm is shown to be efficient while being computationally simpler than the ml or the bartlett processors the disadvantage of the algorithm is that its snr threshold occurs in lower snr than in the ml algorithm

optimal polynomial decay of functions and operator semigroups we characterize the polynomial decay of orbits of hilbert space c 0 semigroups in resolvent terms we also show that results of the same type for general banach space semigroups and functions obtained recently in batty and duyckaerts j evol eq 8 765 780 2008 are sharp this settles a conjecture posed in batty and duyckaerts 2008

the non analytic growth bound of a c0 semigroup and inhomogeneous cauchy problems abstract the non analytic growth bound t of a c0 semigroup t measures the extent to which t can be approximated by a holomorphic function and it is related to spectral properties of the generator a in regions of c far from the real axis we show that t can be characterised by means of fourier multiplier properties of the resolvent of a far from the real axis and also by existence and uniqueness of mild solutions of inhomogeneous cauchy problems of the form u t au t f t on r where the carleman spectra of f and u are far from the origin the corresponding results for the exponential growth bound 0 t have been established earlier by other authors

the spatial operator in the incompressible navier stokes oseen and stokes equations abstract we investigate the spatial operator in the incompressible navier stokes oseen and stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions the theoretical results are derived for a general form of energy stable boundary conditions and applied to a few commonly used ones the analysis is done on a system that simultaneously covers the nonlinear incompressible navier stokes the oseen and the stokes equations when the spectrum of the spatial operator is investigated we restrict the analysis to the oseen and stokes equations the continuous analysis carries over to the discrete setting by using the summation by parts framework

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

wave evolution reflection and transmission along inhomogeneous waveguides abstract one dimensional wave propagation along an inhomogeneous waveguide such as an acoustic horn a plate strip a rod or a beam is considered the equations of motion of a generic waveguide are written in first order form and the properties of the position dependent eigenvalues and eigenvectors of the system are deduced by considering the symmetry and or the energetics of the system the equations of motion are then transformed to wave co ordinates based on the right eigenvectors of the system and a perturbation method is employed to study wave propagation along deterministic periodic and random waveguides attention is then turned to wave reflection at a cut off cross section and a general result for the phase of the reflection coefficient is derived the general theory is illustrated by application to rod and beam examples the method can be likened to a coupled modes approach in which the properties of the eigenvectors or modes arising from symmetry and or energetics are exploited