a transformation of the environment eliminates parabolic equation phase errors it is shown that phase errors associated with the standard parabolic wave equation can be eliminated in a range independent environment by appropriately transforming the environment solutions to the standard parabolic equation pe in the transformed environment are close to solutions to the helmholtz equation in the original environment in the sense that ray and asymptotic mode contributions to the pe wave fields have no phase errors relative to their helmholtz equation counterparts although phase errors are eliminated in the pe wave fields complete asymptotic equivalence to the helmholtz equation wave fields is not attained ray and mode based wave field expansions are used to illustrate similarities and differences between pe and helmholtz equation wave fields numerical simulations show excellent agreement between pe wave fields in the transformed environment and mode based solutions to the helmholtz equation in the true environment

a basic convergence result for particle filtering the basic nonlinear filtering problem for dynamical systems is considered approximating the optimal filter estimate by particle filter methods has become perhaps the most common and useful method in recent years many variants of particle filters have been suggested and there is an extensive literature on the theoretical aspects of the quality of the approximation still a clear cut result that the approximate solution for unbounded functions converges to the true optimal estimate as the number of particles tends to infinity seems to be lacking it is the purpose of this contribution to give such a basic convergence result for a rather general class of unbounded functions furthermore a general framework including many of the particle filter algorithms as special cases is given

a three dimensional parabolic equation model that includes the effects of rough boundaries a three dimensional parabolic equation 3dpe model that handles wide propagation angles in depth narrow propagation angles in azimuth and rough boundaries is derived and solved numerically with the method of alternating directions a homogeneous boundary condition which is easily incorporated into the numerical solution of the 3dpe is applied at the ocean surface to approximate the effects of rough boundaries in terms of a reflection coefficient that depends on grazing angle calculations are presented to demonstrate the accuracy of the 3dpe and to demonstrate that horizontal coupling the effects due to the term involving azimuth derivatives can be important in shallow water previous work has focused on deep water applications the model is applied to problems involving rough boundaries and range dependence

modeling three dimensional propagation in an oceanic wedge using parabolic equation methods in this paper the predictions of azimuthally uncoupled and azimuthally coupled parabolic equation pe methods for three dimensional acoustic propagation in a penetrable oceanic wedge are compared the individual modes of the waveguide at the source position are used as the initial field for the pe methods and the resulting pe fields are compared to the ray diagrams of three dimensional adiabatic mode theory

an optimal pe type wave equation a one way wave equation is presented with the following properties 1 for low angles and small sound speed variations it reduces to the standard parabolic approximation 2 it allows a split step solution 3 the rays associated with this equation are exactly the rays of the helmholtz equation in a range independent environment it is in the last sense an optimal one way wave equation results of the split step solution of this equation are presented and compared to normal mode calculations and results of another modification of the standard parabolic equation which was given by thomson and chapman d j thomson and n r chapman j acoust soc am 74 1848 1854 1983

partial non gaussian state space summary in this paper we suggest the use of simulation techniques to extend the applicability of the usual gaussian state space filtering and smoothing techniques to a class of nongaussian time series models this allows a fully bayesian or maximum likelihood analysis of some interesting models including outlier models discrete markov chain components multiplicative models and stochastic variance models finally we discuss at some length the use of a non gaussian model to seasonally adjust the published money supply figures

a family of symplectic integrators stability accuracy and molecular dynamics applications the following integration methods for special second order ordinary differential equations are studied leapfrog implicit midpoint trapezoid stormer verlet and cowell numerov we show that all are members or equivalent to members of a one parameter family of schemes some methods have more than one common form and we discuss a systematic enumeration of these forms we also present a stability and accuracy analysis based on the idea of modified equations and a proof of symplecticness it follows that cowell numerov and lim2 a method proposed by zhang and schlick are symplectic a different interpretation of the values used by these integrators leads to higher accuracy and better energy conservation hence we suggest that the straightforward analysis of energy conservation is misleading

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

equilibria of runge kutta methods it is known that certain runge kutta methods share the property that in a constant step implementation if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system such methods are calledregular in the present paper we provide a recursive test to check whether given method is regular moreover by examining solution trajectories of linear equations we prove that the order of ans stage regular method may not exceed 2 s 2 2 and that the maximal order of regular runge kutta method with an irreducible stability function is 4

dissipativity of θ methods for nonlinear volterra delay integro differential equations this paper deals with the analytic and numerical dissipativity of nonlinear volterra delay integro differential equations vdides a dissipativity criteria for vdides is given we investigate the dissipativity properties of one leg q methods and linear q methods for the underlying systems it is shown that for q 12 1 any one leg q method and linear q method can inherit the dissipativity of the underlying systems

finite element approximation of the cahn hilliard equation on surfaces abstract in this paper we consider the phase separation on general surfaces by solving the nonlinear cahn hilliard equation using a finite element method a fully discrete approximation scheme is introduced and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time this in turn leads to convergence and optimal error estimates of the discretization scheme numerical examples are also provided to substantiate the theoretical results

dissipativity of θ methods for a class of nonlinear neutral delay differential equations abstract this paper is concerned with the dissipativity of theoretical solution and numerical solution of a class of nonlinear neutral delay differential equations at first a sufficient condition for the dissipativity of theoretical solution of the mentioned problem above is given then the dissipativity results are obtained for one leg and linear methods when they are applied to these problems

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

a two way parabolic equation for acoustic backscattering in the ocean the parabolic equation pe method is generalized to handle backscattered acoustic energy in the ocean the two way pe is based on the single scattering approximation and the approach of two way coupled modes in which range dependent environments are approximated by a sequence of range independent regions at the vertical boundaries between regions the solution of the two way pe is required to satisfy two continuity conditions the range derivative in one of the conditions is replaced by a higher order pe depth operator the reflected and transmitted fields that satisfy these conditions are computed with an efficient iteration scheme the outgoing and incoming fields are propagated by two way range marching the two way pe which is presently implemented for two dimensional problems is a practical method for solving large scale reverberation problems the accuracy of the two way pe is demonstrated by comparisons with reference solutions the two way pe is applied to simulate the localization of a source of backscattering using the method of back propagation

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

mode functions for the wide angle approximation to the parabolic equation the parabolic approximation to the wave equation is examined within the context of normal mode theory in a layered waveguide the horizontal propagation constants and modal amplitudes of the field satisfying the standard parabolic equation spe approximation can be mapped exactly into the amplitudes and wave numbers of the normal modes for the field p satisfying the helmholtz wave equation however this is not the case for certain other parabolic approximations such as the wide angle parabolic equation wape approximation approximate mode functions for the wape approximation are developed these mode functions are then used to decompose range independent sound pressure fields computed using the wape approximation the resulting modal coefficients and eigenfunctions obtained using the wape mode functions are compared with those obtained using standard normal mode theory

the effective stability of adaptive timestepping ode solvers we consider the behavior of certain adaptive timestepping methods based upon embedded explicit runge kutta pairs when applied to dissipative odes it has been observed numerically that the standard local error controls can impart desirable stability properties but this has been rigorously verified only for very special low order runge kutta pairs the rooted tree expansion of a certain quadratic form central to the stability theory of runge kutta methods is derived this together with key assumptions on the sequence of accepted time steps and the local error estimate provides a general explanation for the observed stability of such algorithms on dissipative problems under these assumptions which are expected to hold for typical numerical trajectories two different results are proved first for a large class of embedded runge kutta pairs of order 1 2 controlled on an error per unit step basis all such numerical trajectories will eventually enter a particular bounded set this occurs for sufficiently small tolerances independent of the initial conditions second for pairs of arbitrary orders p 1 p operating under either error per step or error per unit step control similar results are obtained when an additional structural assumption that should be valid for many cases of interest is imposed on the dissipative vector field numerical results support both the analysis and the assumptions made

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

dynamic conditional independence models and markov chain monte carlo methods abstract in dynamic statistical modeling situations observations arise sequentially causing the model to expand by progressive incorporation of new data items and new unknown parameters for example in clinical monitoring patients and data arrive sequentially and new patient specific parameters are introduced with each new patient markov chain monte carlo mcmc might be used for continuous updating of the evolving posterior distribution but would need to be restarted from scratch at each expansion stage thus mcmc methods are often too slow for real time inference in dynamic contexts by combining mcmc with importance resampling we show how real time sequential updating of posterior distributions can be effected the proposed dynamic sampling algorithms use posterior samples from previous updating stages and exploit conditional independence between groups of parameters to allow samples of parameters no longer of interest to be discarded such as when a patient dies or is discharged we apply the

properties of runge kutta summation by parts methods abstract we review and extend the list of stability and convergence properties satisfied by runge kutta rk methods that are associated with summation by parts sbp operators herein called rk sbp methods the analysis covers classical generalized as well as upwind sbp operators previous work on the topic has relied predominantly on energy estimates in contrast we derive all results using a purely algebraic approach that rests on the well established theory of rk methods the purpose of this paper is to provide a bottom up overview of stability and convergence results for linear and non linear problems that relate to general rk sbp methods to this end we focus on the rk viewpoint since this perspective so far is largely unexplored this approach allows us to derive all results as simple consequences of the properties of sbp methods combined with well known results from rk theory in this way new proofs of known results such as a l and b stability are given additionally we establish previously unreported results such as strong s stability dissipative stability and stiff accuracy of certain rk sbp methods further it is shown that a subset of methods are b convergent for strictly contractive non linear problems and convergent for non linear problems that are both contractive and dissipative

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

dissipativity of linear θ methods for integro differential equations this paper is concerned with the analytic and numerical dissipativity of nonlinear integro differential equations ides a dissipativity criteria for ides is given it is shown that any a stable linear 0 method for the systems is dissipative numerical examples are given to confirm the theoretical results

leading order cross term correction of three dimensional parabolic equation models the issue of handling a leading order cross multiplied term in three dimensional 3d parabolic equation pe based models is addressed in particular numerical results obtained incorporating a leading order cross term correction in an existing 3d pe model written in cylindrical coordinates based on higher order pade approximations in both depth and azimuth and a splitting operator technique are reported note that the numerical algorithm proposed in this paper could be used in the future to update any 3d pe codes that neglect cross terms and use a splitting numerical technique the 3d penetrable wedge benchmark problem is chosen to illustrate the accuracy of the now fully wide angle enhanced 3d pe model the comparisons with a 3d reference solution based on the image source clearly show that handling the leading order cross term in the 3d pe computation is sufficient to remove the phase errors inherent to any 3d pe models that neglect cross terms in their formulations

unconditionally stable methods for gradient flow using convex splitting runge kutta scheme abstract we propose a convex splitting runge kutta csrk scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner the key feature of the scheme is a combination of a convex splitting method and a specially designed multi stage two additive runge kutta method our methods are high order accurate in time and assure the gradient energy stability for any time step size we provide detailed proof of the unconditional energy stability and present issues on the practical implementations we demonstrate the accuracy and stability of the proposed methods using numerical experiments of the cahn hilliard equation

a three dimensional propagation algorithm using finite azimuthal aperture the use of three dimensional 3 d propagation models is becoming increasingly common in current underwater acoustics applications such models typically treat the propagation as a function of range depth and bearing consistent with previous multibearing two dimensional n 2 d models however to obtain accurate 3 d solutions at long ranges many bearings must be computed in order to maintain the necessary cross range resolution between bearings in this paper a 3 d parabolic equation pe model is developed with a marching algorithm based on the split step fourier technique in both depth and azimuthal bearing the algorithm includes a scheme to compute the solution for only a finite azimuthal aperture rather than the full 360 of bearing the success of this algorithm allows for increased cross range resolution at long range without increasing the number of bearings needed in the calculation results of this technique are compared to results from a suggested benchmark case

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

bayesian inference and prediction for mean and variance shifts in autoregressive time series abstract this article is concerned with statistical inference and prediction of mean and variance changes in an autoregressive time series we first extend the analysis of random mean shift models to random variance shift models we then consider a method for predicting when a shift is about to occur this involves appending to the autoregressive model a probit model for the probability that a shift occurs given a chosen set of explanatory variables the basic computational tool we use in the proposed analysis is the gibbs sampler for illustration we apply the analysis to several examples

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

numerical solutions of range dependent benchmark problems in ocean acoustics accurate numerical solutions are presented for benchmark problems associated with propagation in a wedge with different bottom boundary conditions as well as in a plane parallel waveguide with changing profile with range the benchmark problems are solved with three different general purpose numerical codes 1 a coupled mode code providing a full spectrum two way backscatter included solution of the wave equation 2 a finite difference pe code providing a wide angle solution of the one way parabolic approximation of the wave equation and 3 a split step pe code providing a more narrow angle solution of the one way parabolic approximation of the wave equation only the two way coupled mode results can be considered accurate solutions of the benchmark problems

uniform ultimate boundedness of numerical solutions to nonlinear neutral delay differential equations this paper is concerned with the long time behaviour of the numerical solutions generated by runge kutta rk methods for nonlinear neutral delay differential equations nddes it is proved that the numerical solutions produced by k l algebraically stable rk methods are uniformly ultimately bounded some examples reveal that some rk methods completely preserve the long time behaviour of the exact solutions to nddes for sufficiently small time stepsize h as a comparison with the previous results a numerical example which further illustrates our theoretical results is provided

a survey of convergence results on particle filtering methods for practitioners optimal filtering problems are ubiquitous in signal processing and related fields except for a restricted class of models the optimal filter does not admit a closed form expression particle filtering methods are a set of flexible and powerful sequential monte carlo methods designed to solve the optimal filtering problem numerically the posterior distribution of the state is approximated by a large set of dirac delta masses samples particles that evolve randomly in time according to the dynamics of the model and the observations the particles are interacting thus classical limit theorems relying on statistically independent samples do not apply in this paper our aim is to present a survey of convergence results on this class of methods to make them accessible to practitioners

a 3 d split step fourier algorithm based on a discrete spectral representation of the propagation equation a propagation method self consistent in the discrete domain for the simulation of large distance scenarios in a 3 d inhomogeneous atmosphere is proposed first a spectral representation of the field in a discrete 3 d domain over an infinite perfect ground is introduced in cylindrical coordinates a discrete spectral propagator is then derived as in 2 d methods the inhomogeneous atmosphere is considered in the spatial domain using phase screens an explicit numerical scheme is introduced for directive sources a sectoral propagation is proposed to reduce the computation time by limiting the computation domain the method is tested by simulation scenarios involving refractive effects in both vertical and azimuthal directions the 3 d refractive effects are shown to be properly simulated

unobserved component time series models with arch disturbances abstract this paper considers how arch effects may be handled in time series models formulated in terms of unobserved components a general model is formulated but this includes as special cases a random walk plus noise model with both disturbances subject to arch effects an arch m model with a time varying parameter and a latent factor model with arch effects in the factors although the model is not conditionally gaussian an approximate filter can be obtained and used as the basis for estimation the performance of this method is examined on real data sets and monte carlo experiments are carried out the method is extended to handle garch and a latent factor model based on the t distribution

asymptotic phase errors in parabolic approximations to the one way helmholtz equation asymptotic in the geometric limit phase errors associated with parabolic approximations to the one way helmholtz equation are investigated in order to estimate the phase accuracy of various parabolic approximations the canonical hamiltonian formalism is used to derive ray equations it is shown that among the class of parabolic approximations that allow the split step fourier algorithm to be used the wide angle c0 insensitive approximation tappert et al j acoust soc am 97 2771 2782 1995 has full second order accuracy other approximations in this class such as the thomson chapman and the berman wright baer approximations are shown to have second order errors as does the standard first order parabolic approximation numerical calculations support the expectation that among this class of parabolic approximations the wide angle c0 insensitive approximation is superior

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

the essential stability of local error control for dynamical systems although most adaptive software for initial value problems is designed with an accuracy requirement control of the local error it is frequently observed that stability is imparted by the adaptation this relationship between local error control and numerical stability is given a firm theoretical underpinning the dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations dissipative contractive and gradient systems dissipative dynamical systems are characterised by having a bounded absorbing set mathcal b which all trajectories eventually enter and remain inside the exponentially contractive problems studied have a unique globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set mathcal b may be chosen to be a ball of arbitrarily small radius around the equilibrium point the gradient systems studied are those for which the set of equilibria comprises isolated points and all traje

equilibrium attractive properties of a class of multistep runge kutta methods the main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep runge kutta methods for initial value problems of ordinary differential equations some algebraic conditions insuring the equilibrium attractivity are given and some methods satisfying these algebraic conditions are constructed some numerical examples confirm our results

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

euler like discrete models of the logistic differential equation abstract to understand the behavior of difference schemes on nonlinear differential equations it seems desirable to extend the standard linear stability theory into a nonlinear theory as a step in that direction we investigate the stability properties of euler related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other we find that these schemes are superior to explicit schemes when they are stable and the blow up time has not passed for these h values they are dynamically faithful when these schemes turn unstable however they have much less desirable properties than explicit or fully implicit schemes they become simultaneously superstable and unstable this is explained by the fact that these schemes are not self adjoint the linearly implicit self adjoint scheme is dynamically faithful in an euler typical range of step sizes and gives correct stability for all step sizes

a higher order energy conserving parabolic equqation for range dependent ocean depth sound speed and density outgoing solutions of the wave equation including parabolic equation pe and normal mode solutions are usually formulated so that pressure is continuous with range for range dependent problems the accuracy of normal mode solutions has been improved by conserving energy rather than maintaining continuity of pressure porter et al the problem of energy conservation in one way equations j acoust soc am 89 1058 1067 1991 this approach is applied to derive a higher order energy conserving pe that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density a special numerical approach and complex pade coefficients are applied to suppress gibbs oscillations the back propagated half space field an improved pe starter is applied to handle wide propagation angles reference solutions generated with a complex ray model and with the rotated pe are used for comparison

dissipativity of runge kutta methods for neutral delay integro differential equations this paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro differential equations the dissipativity results are obtained for algebraically stable runge kutta methods when they are applied to above problems

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

energy stable compact scheme for cahn hilliard equation with periodic boundary condition abstract we present a compact scheme to solve the cahn hilliard equation with a periodic boundary condition which is fourth order accurate in space we introduce schemes for two and three dimensions which are derived from the one dimensional compact stencil the energy stability is completely proven for the proposed scheme based on the application of the compact method and well known convex splitting methods detailed proofs of the mass conservation and unique solvability are also established numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

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

on the use of higher order azimuthal schemes in 3 d pe modeling in this paper the issue of using higher order finite difference schemes to handle the azimuthal derivative term in a three dimensional parabolic equation based model is addressed the three dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes both point source and modal initializations of the pressure field are considered for each higher order finite difference scheme used in azimuth the convergence of the numerical solution with respect to the azimuth is investigated and the cpu times are given some comparisons with solutions obtained from another 3 d model j a fawcett j acoust soc am 93 2627 2632 1993 are presented the numerical simulations show that the use of a higher order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions the higher order schemes have approximately the same efficiency as a fft based approach in fact may outperform it slightly however the finite difference approach has the advantage that it may be more flexible than the fft approach for various pe approximations

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

trust region algorithms and timestep selection unconstrained optimization problems are closely related to systems of ordinary differential equations odes with gradient structure in this work we prove results that apply to both areas we analyze the convergence properties of a trust region or levenberg marquardt algorithm for optimization the algorithm may also be regarded as a linearized implicit euler method with adaptive timestep for gradient odes from the optimization viewpoint the algorithm is driven directly by the levenberg marquardt parameter rather than the trust region radius this approach is discussed for example in r fletcher practical methods of optimization 2nd ed john wiley new york 1987 but no convergence theory is developed we give a rigorous error analysis for the algorithm establishing global convergence and an unusual extremely rapid type of superlinear convergence the precise form of superlinear convergence is exhibited the ratio of successive displacements from the limit point is bounded above and below by geometrically decreasing sequences we also show how an inexpensive change to the algorithm leads to quadratic convergence from the ode viewpoint this work contributes to the theory of gradient stability by presenting an algorithm that reproduces the correct global dynamics and gives very rapid local convergence to a stable steady state

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

structure preservation for constrained dynamics with super partitioned additive runge kutta methods a broad class of partitioned differential equations with possible algebraic constraints is considered including hamiltonian and mechanical systems with holonomic constraints for mechanical systems a formulation eliminating the coriolis forces and closely related to the euler lagrange equations is presented a new class of integrators is defined the super partitioned additive runge kutta spark methods this class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms a linear stability and convergence analysis of these methods is given spark methods allowing the direct preservation of certain properties are characterized different structures and invariants are considered the manifold of constraints symplecticness reversibility contractivity dilatation energy momentum and quadratic invariants with respect to linear stability and structure preservation the class of s stage lobatto iiia b c c spark methods is of special interest controllable numerical damping can be introduced by the use of additional parameters some issues related to the implementation of a reversible variable stepsize strategy are discussed

an energy stable runge kutta method for convex gradient problems abstract we propose a class of runge kutta methods which provide a simple unified framework to solve the gradient flow of a convex functional in an unconditionally energy stable manner stiffly accurate runge kutta methods are high order accurate in terms of time and also assure the energy stability for any time step size when they satisfy the positive definite condition we provide a detailed proof of the unconditional energy stability as well as unique solvability of the proposed scheme we demonstrate the accuracy and stability of the proposed methods using numerical experiments for a specific example

convergence of the markov chain distributed particle filter mcdpf the markov chain distributed particle filter mcdpf is an algorithm for the nodes in a sensor network to cooperatively run a particle filter based on each sensor making updates to a local particle set using only local measurements and then having particles exchanged between neighboring sensors based on a markov chain on the network this paper extends previously known almost sure convergence results for the mcdpf to prove that the mcdpf convergences to the optimal filter in mean square as the number of particles and the number of markov chain steps both go to infinity the convergence proof derives an explicit error bound showing that the convergence is inverse square root in both parameters a numerical example is provided to support the theoretical result