a chebyshev spectral method for gas transients in pipelines a chebyshev pseudospectral method is proposed for slow transients simulation of gas transportation systems the scheme is theoretically investigated comparisons with the lax wendroff scheme prove that the chebyshev method is more efficient in terms of storage and computer time infinite order accuracy for smooth solutions is predicted by the theory and observed in the experiments

almost periodic solutions of forced vectorial lienard equations abstract we describe the set of bounded or almost periodic solutions of the following lienard system u d dt f u cu e t where e r r n is almost periodic c r n r n is a symmetric and nonsingular linear operator and f denotes the gradient of the convex function f on r n then we state a result of existence and uniqueness of almost periodic solution

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

accurate global solutions of em boundary value problems for coaxial radiators we present a new numerical method called the field domain partition combining solution for the analysis of complicated electromagnetic em boundary value problems bvp this is a numerical procedure based on the combination of the global numerical boundary condition gnbc and the finite element method using this method a variety of coaxial radiators can be precisely treated as bvp s and rigorously solved for their accurate global solution three examples are given demonstrating that the present numerical procedure is very general flexible and effective for both theoretical and engineering purposes

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

properties of some weighted sobolev spaces and application to spectral approximations the weighted sobolev spaces on a square whose weight is a given power of the product of the distances to the edges are introduced trace theorems related to these spaces are proved then a regularity result for the dirichlet problem for the laplace operator is proved next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated finally a collocation spectral method for the dirichlet problem for the laplace operator with inhomogeneous boundary conditions is analyzed this work includes new results on chebyshev approximation

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

adaptive spectral methods with application to mixing layer computations this paper reports some experiments on the use of adaptive chebyshev pseudospectral methods for compressible mixing layer computations different functionals measuring the optimality of the polynomial approximation are discussed and compared in particular we address the problem of the practical computation of the various functionals the utility of the self adaptive method is then demonstrated by some examples from compressible mixing layer calculations

filtered stochastic galerkin methods for hyperbolic equations abstract uncertainty quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks standard intrusive methods such as stochastic galerkin sg lead to oscillatory solutions and can result in non hyperbolic moment systems the intrusive polynomial moment ipm method guarantees hyperbolicity but comes at higher numerical costs in this paper we filter the generalized polynomial chaos gpc coefficients of the sg approximation which allows a numerically cheap reduction of oscillations the derived filter is based on lasso regression which sets small gpc coefficients of high order to zero we adaptively and automatically choose the filter strength to obtain a zero valued highest order moment the filtered sg method is tested for burgers and the euler equations results show a reduction of oscillations at shocks which leads to an improved approximation of expectation values and the variance compared to sg and ipm

analysis of the kleiser schumann method the kleiser schumann algorithm for the approximation of the stokes problem by fourier legendre polynomials is analized stability when the degree of the polynomials increases is established whereas error estimates in sobolev spaces are proven

domain decomposition methods for pseudo spectral approximations part i second order equations in one dimension multidomain pseudo spectral approximations of second order boundary value problems in one dimension are considered the equation is collocated at the chebyshev nodes inside each subinterval different patching conditions at the interfaces are analyzed results of stability and convergence are given

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

the stability of pseudospectral chebyshev methods abstract the stability of pseudospectral chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients the choice of collocation points is discussed numerical examples are given for the case of variable coefficient hyperbolic equations author

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

positive filtered p _n moment closures for linear kinetic equations we propose a positive preserving moment closure for linear kinetic transport equations based on a filtered spherical harmonic fp n expansion in the angular variable the recently proposed fp n moment equations are known to suffer from the occurrence of unphysical negative particle concentrations the origin of this problem is that the fp n approximation is not always positive at the kinetic level the new fp n closure is developed to address this issue a new spherical harmonic expansion is computed via the solution of an optimization problem with constraints that enforce positivity but only on a finite set of preselected points combined with an appropriate pde solver for the moment equations this ensures positivity of the particle concentration at each step in the time integration under an additional mild regularity assumption we prove that fp n has the same consistency as fp n that is the fp n approximation converges to a given target function in l 2 at the same rate a

pseudo almost automorphic mild solutions to semilinear integral equations in a banach space abstract in this paper we consider the existence of pseudo almost automorphic solutions of the semilinear integral equation x t t a t s a x s f s x s d s t r in a banach space x where a l 1 r a is the generator of an integral resolvent family of linear bounded operators defined on the banach space x and f r x x is a pseudo almost automorphic function the main results are proved by using integral resolvent families combined with the theory of fixed points

on the boundary treatment in spectral methods for hyperbolic systems abstract spectral methods have been successfully applied to the simulation of slow transients in gas transportation networks implicit time advancing techniques are naturally suggested by the nature of the problem the aim of this paper is to clarify the correct treatment of the boundary conditions in order to avoid any stability restriction originated by the boundaries the beam and warming and the lerat schemes are unconditionally linearly stable when used with a chebyshev pseudospectral method engineering accuracy for a gas transportation problem is achieved at courant numbers up to 100

combined finite element and spectral approximation of the navier stokes equations we present a method for the numerical approximation of navier stokes equations with one direction of periodicity in this direction a fourier pseudospectral method is used in the two others a standard f e m is applied we prove optimal rate of convergence where the two parameters of discretization intervene independently on presente une methode d approximation numerique des equations de navier stokes possedant une direction de periodicite dans cette direction une methode pseudospectrale basee sur des developpements en serie de fourier est utilisee dans les deux autres on applique une methode d elements finis standard on montre que la convergence est optimale et que les deux parametres de discretisation peuvent etre choisis de facon independante

almost periodic solutions of second order systems with monotone fields on a compact subset we give sufficient conditions for the existence of almost periodic solutions of the second order differential system u f t u when f c0 r k rn with k is a compact convex subset of rn and the partial function f t is monotone for each t r for existence we do not assume that f t is strictly monotone

almost periodic solutions for abstract impulsive differential equations this paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in banach space the investigations are carried out by means of the fractional powers of operators we construct an example to illustrate the feasibility of our results

fourier spectral approximation to long time behaviour of the derivative three dimensional ginzburg landau equation in this paper we consider a derivative ginzburg landau equation with periodic initial value condition in three dimensional space a fully discrete galerkin fourier spectral approximation scheme is constructed and then the dynamical behaviour of the discrete system is analysed firstly the existence of global attractors an of the discrete system are proved by a priori estimate of the discrete solution next the convergence of approximate attractors is proved by error estimates of the discrete solution furthermore the long time convergence as n and 0 simultaneously as well as the numerical long time stability of the discrete scheme are obtained

almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in banach spaces in this paper we prove some existence results for almost automorphic and pseudo almost automorphic mild solutions to a class of abstract differential equations in banach spaces the main technique is based on some composition theorems combined with the contraction mapping theorem finally we present an application to a semilinear partial differential equation with dirichlet conditions

multi domain parallel computation of wake flows abstract we present a new multi domain parallel computational method for simulation of unsteady flows involving a primary object a long wake region and possibly a secondary object affected by the wake flow the method is based on the stabilized finite element formulation of the time dependent navier stokes equations of incompressible flows in the multi domain computational method the entire simulation domain is divided into an ordered sequence of overlapping subdomains the flow data computed over the leading subdomain is used for specifying the inflow boundary conditions for the next subdomain the subdomain corresponding to the wake would not involve any objects hence the mesh constructed over this domain would be structured a special purpose finite element implementation for structured meshes is used for the wake domain to achieve much higher computational speeds compared to a general purpose implementation we present verification studies for the multi domain method and special purpose implementation followed by two numerical examples the first example is the wake behavior behind a circular cylinder the second one is the aerodynamic effect of tip vortices released from a leading wing on a trailing wing placed in the far wake

spectral tau approximation of the two dimensional stokes problem we analyse the spectral tau method for the approximation of the stokes system on a square with dirichlet boundary conditions we provide an error estimate in the norm of the sobolev spaceh s for the approximation of a divergence free vector field with polynomial divergence free vector fields we apply this result to prove some convergence estimates for the solution of the discrete stokes problem

crank nicolson and legendre spectral collocation methods for a partial integro differential equation with a singular kernel abstract in this paper we present an efficient numerical method for the solution of a partial integro differential equation with a singular kernel in the time direction a crank nicolson finite difference scheme is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term also for space discretization we apply legendre spectral collocation method we discuss the stability and convergence of proposed method and show that the method is unconditionally stable and convergent with order o 3 2 n s where n and s are time step size number of collocation points and regularity of exact solution respectively we compare the numerical results of proposed method with the results of other schemes in the literature in terms of accuracy computational order and cpu time to show the efficiency and applicability of it

approximation of infinite delay and volterra type equations we consider a class of infinite delay equations of neutral type which includes volterrra type integral und integrodifferential equations using abstract approximation results cf trotter kato type for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by laguerre and legendre polynomials numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data

a hybrid analytical numerical technique for elliptic pdes recent work has given rise to a novel and simple numerical technique for solving elliptic boundary value problems formulated in convex polygons in two dimensions the method based on the unified t

bounded and almost automorphic solutions of some nonlinear differential equations in banach spaces abstract we study some properties of bounded and almost automorphic solutions of the following nonlinear differential equation x f t x where x is a banach space and f r x x is almost automorphic with a parameter we give sufficient conditions for the existence of almost automorphic solutions when the differential equation admits at least a solution with a relatively compact range defined on the half line of positive numbers

almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional applications to heat and wave equations with nonlinearities in this work we study the existence of bounded and almost automorphic solutions for evolution equations in banach spaces we suppose that the linear part is the infinitesimal generator of a compact c0 semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument we give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on r we use the subvariant functional method to show that every k minimizing mild solution is compact almost automorphic applications are provided for both heat and wave equations with nonlinearities in several functional spaces

stabilization of spectral methods by finite element bubble functions abstract we show that a standard spectral collocation scheme can be stabilized by adding extra trial test functions with local support bubbles applications are given to a scalar advection diffusion problem and to the stokes problem

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

long time stability of high order multistep numerical schemes for two dimensional incompressible navier stokes equations the long time stability properties of a few multistep numerical schemes for the two dimensional incompressible navier stokes equations formulated in vorticity stream function are investigated in this article these semi implicit numerical schemes use a combination of explicit adams bashforth extrapolation for the nonlinear convection term and implicit adams moulton interpolation for the viscous diffusion term up to fourth order accuracy in time as a result only two poisson solvers are needed at each time step to achieve the desired temporal accuracy the fully discrete schemes with fourier pseudospectral approximation in space are analyzed in detail with the help of a priori analysis and aliasing error control techniques we prove uniform in time bounds for these high order schemes in both l 2 and h m norms for m ge 1 provided that the time step is bounded by a given constant such a long time stability is also demonstrated by the numerical experiments

on the existence of global smooth solutions for a model equation for fluid flow in a pipe abstract we model the flow of a fluid in a pipe by a first order nonlinear hyperbolic system with zero order nonlinear dissipation we prove that a unique global smooth solution exists if the initial data are in an appropriate invariant region and if the first derivatives of the initial data are sufficiently small

spectral approximation of pattern forming nonlinear evolution equations with double well potentials of quadratic growth this paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as l 2 gradient flow for the modica mortola regularization of the functional v bv t d 1 1 e v 2 td v 1 2 k d k v k 2 here is the interfacial energy per unit length or unit area t d is the flat torus in d and is a nonnegative fourier multiplier that is continuous on d symmetric in the sense that for all d and that decays to zero at infinity such functionals feature in mathematical models of pattern formation in micromagnetics and models of diblock copolymers the resulting evolution equation is discretized by a fourier spectral method with respect to the spatial variables and a modified crank nicolson scheme in time optimal order a priori bounds are derived on the global error in the l 0 t l 2 t d norm

accurate computation of the radiation from simple antennas using the finite difference time domain method two antennas are considered a cylindrical monopole and a conical monopole both are driven through an image plane from a coaxial transmission line each of these antennas corresponds to a well posed theoretical electromagnetic boundary value problem and a realizable experimental model these antennas are analyzed by a straightforward application of the finite difference time domain fd td method the computed results for these antennas are shown to be in excellent agreement with accurate experimental measurements for both the time domain and the frequency domain the graphical displays presented for the transient near zone and far zone radiation from these antennas provide physical insight into the radiation process

global numerical boundary condition based pde solution techniques for open region electromagnetic field problems the underlying concept and algorithm for global numerical boundary condition gnbc based pde solution techniques for open region emf problems are presented this numerical procedure has been applied with success for various boundary value problems in scattering waveguide and radiation analysis showing distinct advantages in generality flexibility and simplicity all the gnbcs treated are of the first kind i e eigenfunction based gnbcs

memory universal prediction of stationary random processes we consider the problem of one step ahead prediction of a real valued stationary strongly mixing random process xi sub i spl infin sup spl infin the best mean square predictor of x sub 0 is its conditional mean given the entire infinite past x sub i sub i spl infin sup 1 given a sequence of observations x sub 1 x sub 2 x sub n we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension for example neural networks and legendre polynomials the proposed estimators select both the model memory and the model dimension in a data driven fashion by minimizing certain complexity regularized least squares criteria when the underlying predictor function has a finite memory we establish that the proposed estimators are memory universal the proposed estimators which do not know the true memory deliver the same statistical performance rates of integrated mean squared error as that delivered by estimators that know the true memory furthermore when the underlying predictor function does not have a finite memory we establish that the estimator based on legendre polynomials is consistent

the approximation theory for the p version of the finite element method the purpose of this article is to present the approximation theory underlying the p version of the finite element method by exploiting the relationship between polynomial approximation and certain weighted sobolev spaces which are identified as the domains of positive real powers of the legendre operator this one dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted sobolev spaces on triangulated domains of r n it is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions inverse results are given for the special case of approximation in l 2

on numerical boundary treatment of hyperbolic systems for finite difference and finite element methods it is well known that the stability of the initial boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment in fact we show that for any boundary treatment one can construct systems for which the boundary treatment on the natural variables is not stable our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation this is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary in a working code this can be accomplished by adding a correction term to the existing boundary algorithm the analysis and computational results are presented for both finite differences and semi discrete galerkin methods

a spectral difference method for two dimensional viscous flow abstract we propose a spectral difference method for the 2 dimensional vorticity equation with a periodic boundary condition in one direction the solution satisfies a semidiscrete conservation law and thus better numerical results are obtained we also prove stability and convergence

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

error estimates for spectral and pseudospectral approximations of hyperbolic equations the linear one dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered we investigate some spectral and pseudospectral collocation approximations of jacobi and legendre type in the latter case the boundary condition is implicitly treated stability and convergence results in weighted norms are given some precise estimates of the rate of convergence in terms of the regularity of the continuous solution are derived

spectral and pseudo spectral approximations of the navier stokes equations galerkin and collocation approximations by trigonometric polynomials to the stationary navier stokes equations with periodic boundary conditions are analyzed stability results and optimal rates of convergence for both velocity and pressure in sobolev spaces are proven essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems brezzi rappag raviart numer math 36 1980 pp 1 25 maday quarteroni rairo numer anal 1980

parameter estimation by spectral approximation in this paper chebyshev and legendre approximations are proposed for estimating parameters in differential equations which are easy to be performed the convergence and the spectral accuracy are proved even without some conditions as imposed in other papers the numerical results show the advantages of this new approach

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

two completely explicit and unconditionally convergent fourier pseudo spectral methods for solving the nonlinear schrodinger equation abstract this paper aims to construct and analyze two new fourier pseudo spectral fps methods for the general nonlinear schrodinger nls equation the two fps methods have two merits unconditional convergence and complete explicitness in the practical computation further more by introducing a modified mass functional and a modified energy functional the two fps methods are proved to preserve the total mass and energy in the discrete sense besides the standard energy method the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique without any restriction on the grid ratio and initial value we establish the optimal error estimate of the two fps methods for solving the general nls equation while previous work just is valid for the cubic nls equation and requires small initial value for the focusing case these two fps methods are proved to be spectrally accurate in space and second order accurate in time respectively the analysis framework can be used to prove the unconditional convergence of many other fourier pseudo spectral methods for solving the nls type equations we investigate the effect of the nonlinear term on the progression simulation of the plane wave the conservation of the invariants and the effect of initial data on the blow up solution via different parameters numerical results are reported to show the accuracy and efficiency of the proposed methods

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

the fourier chebyshev spectral method for solving two dimensional unsteady vorticity equations abstract we propose a mixed method for solving the two dimensional unsteady vorticity equations by using the fourier spectral approximation in the periodic direction and chebyshev spectral approximation in the non periodic direction some numerical results are given which are compared with those of other methods stability of the scheme and the optimal rate of convergence are proved

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution