two grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

backstepping with bounded feedbacks for time varying systems a family of time varying nonlinear systems is globally uniformly asymptotically stabilized by bounded feedbacks constructed through a new extension of the backstepping approach explicit expressions of control laws and lyapunov functions are given

approximation of biholomorphic mappings by automorphisms of cn in this paper we prove several results on approximation of biholomorphic mappings between domains in c n 2 by holomorphic automorphisms of c recall that the group of holomorphic automorphisms of c denoted aut c consists of those holomorphic mappings c c that have a holomorphic inverse 4 1 c c with the topology of uniform convergence on compact sets aut c is a topological group while the group aut c 1 of automorphisms of the complex plane consists only of the linear mappings az b a b e c a 4 0 the group aut c is very large and complicated when n 2 let us choose a linear coordinate system on c and write the coordinates as z z w where z z1 zn i c c n 1 and w z c then autc n contains mappings of the form

algebraic volume density property of affine algebraic manifolds we introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it in particular that it implies the volume density property the main results of the paper are producing two big classes of examples of stein manifolds with volume density property one class consists of certain affine modifications of n equipped with a canonical volume form the other is the class of all linear algebraic groups equipped with the left invariant volume form

finite difference schemes for the parabolic equation in a variable depth environment with a rigid bottom boundary condition we consider a linear schrodinger type partial differential equation the parabolic equation of underwater acoustics in a layer of water bounded below by a rigid bottom of variable topography using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori h1 estimate and prove an optimal order error bound in the maximum norm for a crank nicolson type finite difference approximation of its solution we also consider the same problem with an alternative rigid bottom boundary condition due to abrahamsson and kreiss and prove again a priori h1 estimates and optimal order error bounds for a crank nicolson scheme

further results on strict lyapunov functions for rapidly time varying nonlinear systems we explicitly construct global strict lyapunov functions for rapidly time varying nonlinear control systems the lyapunov functions we construct are expressed in terms of oftentimes more readily available lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable this leads to new sufficient conditions for uniform global exponential uniform global asymptotic and input to state stability of fast time varying dynamics we also construct strict lyapunov functions for our systems using a strictification approach we illustrate our results using several examples

infinite dimensionality of the automorphism groups of homogeneous stein manifolds we show that the group rm aut mathcal o x of holomorphic automorphisms of a stein manifold x with dim x 2 is infinite dimensional provided x is a homogeneous space of a holomorphic action of a complex lie group

conservative initial boundary value problems for the wide angle pe in waveguides with variable bottoms we consider the third order claerbout type wide angle parabolic equation pe in the context of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom b of range dependent topography there are strong indications that the initial boundary value problem for this equation with just a homogeneous dirichlet boundary condition on b may not be well posed for example when b is downsloping in previous work we proposed an additional bottom boundary condition that together with the zero field condition on b yields a well posed problem in the present paper we continue our investigation of additional bottom boundary conditions that yield well posed physically correct problems motivated by the fact that the solution of the wide angle pe in a domain with horizontal layers conserves its l2 norm in the absence of attenuation we seek additional boundary conditions on a variable topography bottom that yield l2 conservative solutions of the problem we identify a family of such boundary conditions after a range dependent change of the depth variable that makes b horizontal we discretize the continuous problems by second order accurate crank nicolson type finite difference schemes and show by means of numerical experiments that the new models yield accurate simulations of the acoustic field in standard wedge type domains with upsloping and downsloping bottoms

global stability results for switched systems based on weak lyapunov functions in this paper we study the stability of nonlinear and time varying switched systems under restricted switching we approach the problem by decomposing the system dynamics into a nominal like part and a perturbation like one most stability results for perturbed systems are based on the use of strong lyapunov functions i e functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state however switched systems under restricted switching may not admit strong lyapunov functions even when asymptotic stability is uniform over the set of switching signals considered the main contribution of the current paper consists in providing stability results that are based on the stability of the nominal like part of the system and require only a weak lyapunov function these results may have wider applicability than results based on strong lyapunov functions the results provided follow two lines first we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal like and the perturbation like parts of the system second we provide input to state stability iss results for the case when the nominal like part is switched linear time varying we provide two types of iss results standard iss that involves the essential supremum norm of the input and a modified iss that involves a power type norm

analysis of an implicit finite difference scheme for a third order partial differential equation in three dimensions abstract am implicit finite difference scheme approximating a third order partial differential equation in three dimensions is examined the scheme is derived shown to be consistent and second order accurate and a discussion of its stability properties is begun the partial differential equation has application in the propagation of sound underwater

reduction model approach for linear time varying systems with delays we study stabilization problems for time varying linear systems with constant input delays our reduction method ensures input to state stability with respect to additive uncertainties under arbitrarily long delays it applies to rapidly time varying systems and gives a lower bound on the admissible rapidness parameters we also cover slowly time varying systems including upper bounds on the allowable slowness parameters we illustrate our work using a pendulum model

superconvergence of a finite element approximation to the solution of a sobolev equation in a single space variable a standard galerkin method for a quasilinear equation of sobolev type using continuous piecewise polynomial spaces is presented and analyzed optimal order error estimates are established in various norms and nodal superconvergence is demonstrated discretization in time by explicit single step methods is discussed

error estimates for finite difference methods for a wide angle parabolic equation we consider a model initial and boundary value problem for a third order partial differential equation pde a wide angle parabolic equation frequently used in underwater acoustics with depth and range dependent coefficients in the presence of horizontal interfaces and dissipation after commenting on the existence uniqueness theory of solution of the equation we discretize the problem by a second order finite difference method of crank nicolson type for which we prove stability and optimal order error estimates in suitable discrete l2 h1 and maximum norms we also prove under certain conditions that the forward euler scheme is also stable and convergent for the problem at hand

criteria for the density property of complex manifolds in this paper we suggest new effective criteria for the density property this enables us to give a trivial proof of the original anders en lempert result and to establish almost free of charge the algebraic density property for all linear algebraic groups whose connected components are different from tori or c as another application of this approach we tackle the question asked among others by f forstneri v c about the density of algebraic vector fields on euclidean space vanishing on a codimension 2 subvariety

boundary conditions for the parabolic equation in a range dependent duct the parabolic wave equation in a range dependent duct with a rigid bottom is considered the range dependence will introduce amplitude errors in addition to the usual phase error of the local modes it is shown that the amplitude error increases by local mode angle and slope of the bottom and is inversely proportional to the depth imposing the vanishing of the normal derivative of the pressure at the bottom which is the physically correct boundary condition might also lead to ill conditioning of the parabolic model various ways to replace the boundary condition u n 0 are discussed and their accuracy is assessed both numerically and theoretically

asymptotic stability equals exponential stability and iss equals finite energy gain if you twist your eyes in this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables the same is shown for input to state stability and input to state exponential stability and for input to state exponential stability and a nonlinear h estimate

galerkin methods for parabolic and schrodinger equations with dynamical boundary conditions and applications to underwater acoustics in this paper we consider galerkin finite element methods that approximate the solutions of initial boundary value problems in one space dimension for parabolic and schrodinger evolution equations with dynamical boundary conditions error estimates of optimal rates of convergence in l 2 and h 1 are proved for the associated semidiscrete and fully discrete crank nicolson galerkin approximations the problem involving the schrodinger equation is motivated by considering the standard parabolic paraxial approximation to the helmholtz equation used in underwater acoustics to model long range sound propagation in the sea in the specific case of a domain with a rigid bottom of variable topography this model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations in the real parabolic case numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

applications and time domain solution of higher order parabolic equations in underwater acoustics a higher order parabolic equation and the corresponding higher order time domain parabolic equation are derived from a pade series and solved numerically these models provide accurate solutions for problems involving very wide angle propagation e g propagation in the nearfield or over a hard ocean bottom propagation in domains in which sound speed variations are large e g propagation in deep water deep within the ocean bottom in high speed ocean bottoms and possibly of different wave types and propagation out to very long ranges the possibility of modeling elastic wave propagation with a similar approach is considered

finite difference methods for the wide angle parabolic equation we consider a model initial and boundary value problem for the wide angle parabolic equation lur icu of underwater acoustics where l is a second order differential operator in the depth variable z with depth and range dependent coefficients we discretize the problem by the crank nicolson finite difference scheme and also by the forward euler method using nonuniform partitions both in depth and in range assuming that the problem admits a smooth solution and l is invertible for all r under the posed boundary and interface conditions we show stability of both schemes and derive error estimates

discontinuous galerkin methods for the linear schrodinger equation in non cylindrical domains the convergence of a discontinuous galerkin method for the linear schrodinger equation in non cylindrical domains of mathbb r m m 1 is analyzed in this paper we show the existence of the resulting approximations and prove stability and error estimates in finite element spaces of general type when m 1 the resulting problem is related to the standard narrow angle parabolic approximation of the helmholtz equation as it appears in underwater acoustics in this case we investigate theoretically and numerically the order of convergence using finite element spaces of piecewise polynomial functions

a finite difference treatment of interface conditions for the parabolic wave equation the irregular interface a finite difference solution to the parabolic wave equation is extended to treat an irregular interface interface conditions are developed to preserve continuity of pressure and continuity of the normal component of particle velocity at the boundary between media having different sound speeds and densities a complete mathematical treatment for the case of an irregular interface is presented to demonstrate the method numerical results are obtained for sound propagation from deep to shallow water

local asymptotic stability via averaging for time varying systems with unbounded dynamics with respect to time the purpose of this note is to provide sufficient conditions for asymptotic stability for time varying systems whose dynamics are unbounded with respect to time an averaging approach is established that constitutes a generalization of the well known methodology adopted in the literature for systems whose dynamics are bounded with respect to time

a new notion of tameness abstract we generalize the notion of tame discrete sets introduced by rosay and rudin from complex euclidean space to arbitrary complex manifolds and establish their basic properties we show that complex linear algebraic groups different from the complex line or the punctured complex line contain tame discrete sets

parabolic wave equation approximations in heterogenous media the properties of different variants of parabolic approximations of scalar wave equations are analyzed these equations are of general form which includes those used in seismology underwater acoustics and other applications a new version of the parabolic approximation is derived for heterogeneous media it has optimal properties with respect to wave reflection at material interfaces the amplitudes of the reflected and transmitted waves depend continuously on the interface existence uniqueness and energy estimates are proved

crank nicolson finite element discretizations for a two dimensional linear schrodinger type equation posed in a noncylindrical domain first published in mathematics of computation online 2014 84 2015 1571 1598 published by the american mathematical society

further remarks on strict input to state stable lyapunov functions for time varying systems we study the stability properties of a class of time varying non linear systems we assume that non strict input to state stable iss lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given lyapunov functions which guarantee the existence of strict iss lyapunov functions for our systems next we provide simple direct constructions of explicit strict iss lyapunov functions for our systems by applying an integral smoothing method we illustrate our constructions using a tracking problem for a rotating rigid body

averaging criteria for asymptotic stability of time varying systems we establish averaging type sufficient conditions for local asymptotic stability for nonlinear time varying systems our main result is based on an extension of the averaging methodology employed in a recent paper by tsinias and stamati published in the same journal dealing with asymptotic stability for slow time varying systems

density property for hypersurfaces uv p bar x we study hypersurfaces of mathbb c n 2 bar x u v given by equations of form uv p bar x where the zero locus of a polynomial p is smooth reduced the main result says that the lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the lie algebra of all algebraic vector fields consequences of this result for some conjectures of affine algebraic geometry and for the oka grauert gromov principle are discussed

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

the rational approximation to the acoustic wave equation with bottom interaction the rational linear approximation to the wave equation is a full wave approach to modeling range dependent ocean acoustic propagation with bottom interaction it is a one way wave equation which gives an accurate treatment of high angle propagation to angles of about 40 with respect to the horizontal reflection from sound speed and density discontinuities is treated using the natural wave equation matching conditions bathymetry is allowed to vary in range a tridiagonal implicit finite difference solution of this equation has been implemented it has several advantages over the tridiagonal crank nicholson solution of the parabolic equation it more accurately models high angles of propagation treats attenuation as a function of path length rather than range and models range dependent bathymetry in a way that suits the form of the one way wave equation the numerical methods are fourth order accurate in depth the resulting implicit range step is still in the simple tridiagonal form

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

the problem of energy conservation in one way models it is shown that the standard stair step representation of a sloping bottom may result in significant prediction errors in fact current parabolic equation implementations are not energy conserving the problem is shown to derive from the approximate treatment of the interface conditions at vertical boundaries along the stair steps several improved interface conditions are proposed

applications of energy methods to finite difference solutions of the parabolic wave equation abstract implicit finite difference techniques may be applied readily to solve acoustic wave propagation problems in diverse ocean environments for only the most simple cases however can the stability of such schemes be established in this paper the method of energy inequalities is applied to examine the stability and step size requirements for solving the parabolic approximation to the wave equation

the time domain solution of the wide angle parabolic equation including the effects of sediment dispersion the wide angle time domain parabolic equation tdpe which is the inverse fourier transform of the wide angle parabolic equation pe is derived a numerical solution for the model is described and a benchmark calculation is presented the narrow angle tdpe is also considered and its error is analyzed and compared with the error of the narrow angle pe the tdpe is compared with the progressive wave equation which is shown to be restricted to narrow angle propagation for practical purposes in the sediment attenuation is assumed to depend linearly on frequency and the corresponding causal dispersion law is assumed the model is used to show that the effect of sediment dispersion on pulse propagation in the ocean can be significant

finite difference discretizations of some initial and boundary value problems with interface we analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation the schrodinger equation and the wave equation by finite difference methods extending the concept of the elliptic projection well known from the analysis of galerkin finite element methods to our finite difference case we prove second order error estimates in space and time in the 12 norm

a wide angle split step algorithm for the parabolic equation in this paper we describe a new wide angle parabolic equation based on an operator splitting that permits the use of a marching type fourier transform solution method the equation was first presented by feit and fleck appl opt 17 3990 3998 1978 for studying propagation within optical fibers existing computer codes which numerically solve the standard parabolic equation of ocean acoustics by the split step algorithm of tappert and hardin are easily modified to accommodate the wide angle capability of the new equation in addition since the new wide angle equation is less sensitive to the value of the reference wavenumber the effects of phase errors are greatly reduced the results of a simple error analysis indicate that improved accuracy can be achieved by the new wide angle equation for propagation conditions typical of deep ocean environments this is supported by our numerical experience a summary of which is presented in the paper for test cases where the variation of the acoustic index

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

analysis of an implicit finite difference solution to an underwater wave propagation problem abstract an implicit finite difference scheme approximating a third order partial differential equation is examined the scheme is derived shown to be consistent and its stability properties are analyzed the partial differential equation is a parabolic approximating equation to the reduced wave equation

a sufficient condition for asymptotic stability for a class of time varying parameterized systems abstract we derive sufficient conditions for local asymptotic stability for a class of parameterized time varying systems whose dynamics are in general unbounded in time our main result generalizes a well known limiting dynamics based approach due to peuteman and aeyels 1999 in 9 for fast time varying systems whose dynamics are bounded in time its proof is based on a new result providing a lyapunov characterization for exponential stability