asymptotic correction of numerov s eigenvalue estimates with natural boundary conditions abstract asymptotic correction at negligible extra cost greatly improves the accuracy of higher eigenvalues computed by finite difference or finite element methods and generally increases the accuracy of the lower ones as well this paper gives a brief overview of the technique and describes how its previous use with numerov s method may be extended to problems with natural boundary conditions numerical results indicate that it is just as successful as with dirichlet boundary conditions

boundary dynamics driven entanglement we will show how to generate entangled states out of unentangled states on a bipartite system by means of dynamical boundary conditions the auxiliary system is defined by a symmetric but not self adjoint hamiltonian and we will also study the space of self adjoint extensions of the bipartite system we will show that only a small set of these extensions leads to separable dynamics and we will characterize these extensions various simple examples illustrating this phenomenon are discussed in particular we will analyze the hybrid system consisting of a planar quantum rotor and a spin system under a wide class of boundary conditions

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

the generalized riemann problem for reactive flows abstract a generalized riemann problem is introduced for the equations of reactive non viscous compressible flow in one space dimension initial data are assumed to be linearly distributed on both sides of a jump discontinuity the resolution of the singularity is studied and the first order variation in time of flow variables is given in exact form

a second order godunov type scheme for compressible fluid dynamics abstract a second order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented two related versions are proposed one lagrangian and the second direct eulerian they both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity assuming initial nonvanishing slopes on both sides while this solution is an extension of the solution to the classical riemann problem the resulting schemes are second order extensions of godunov s methods in both cases they are very simple to implement in computer codes several numerical examples are shown where the only additional mechanism is a simple monotonicity algorithm

asymptotic stability for a linear difference equation with variable delay we shall obtain sufficient conditions for the uniform stability and the global asymptotic stability of the linear difference equation with variable delay xn 1 xn pnxn kn 0 n n where pn is a sequence of nonnegative real numbers kn is a sequence of nonnegative integers and there exists a nonnegative integer k such that kn k for n n

a laplace operator with boundary conditions singular at one point we discuss a recent paper of berry and dennis j phys a math theor 2008 41 135203 concerning a laplace operator on a smooth domain with singular boundary condition we explain a paradox in the article j phys a math theor 2008 41 135203 and show that if a certain additional condition is imposed the result is a spectral problem for a self adjoint operator having only eigenvalues and no continuous spectrum the eigenvalues accumulate at only and we obtain the asymptotic behaviours of the counting functions n and n for positive and negative eigenvalues the physical meaning of the additional boundary condition is not yet clear

numerical solutions of the spectral problem for arbitrary self adjoint extensions of the one dimensional schrodinger equation a numerical algorithm to solve the spectral problem for arbitrary self adjoint extensions of one dimensional regular schrodinger operators is presented it is shown that the set of all self adjoint extensions of one dimensional regular schrodinger operators is in one to one correspondence with the group of unitary operators on the finite dimensional hilbert space of boundary data and they are characterized by a generalized class of boundary conditions that include the well known dirichlet neumann robin and quasi periodic boundary conditions the numerical algorithm is based on a nonlocal boundary extension of the finite element method and their convergence is proved an appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions and the conditioning of its computation is analyzed some significant numerical experiments are also discussed as well as the comparison with some standard algorithms in particular it is shown that appropriate perturbations of stand

asymptotic behavior of solutions for neutral difference equations the boundedness and asymptotic behavior of all solutions of the neutral delay difference equation of the form d x n c nx n m p nx n k q nx n l 0 n 0 1 2 are investigated where k l m are positive integers c n q n and p n are three sequences of real numbers the obtained sufficient conditions extend and improve the existing results in the literature

a direct eulerian grp scheme for relativistic hydrodynamics two dimensional case this paper develops a direct eulerian generalized riemann problem grp scheme for two dimensional 2d relativistic hydrodynamics rhd it is an extension of the grp scheme for one dimensional 1d rhds z c yang p he h z tang j comput phys 230 2011 7964 7987 and the grp scheme for the non relativistic hydrodynamics m ben artzi j q li g warnecke j comput phys 218 2006 19 43 in order to derive the direct eulerian grp scheme the local grp of the split 2d rhd equations in the eulerian formulation has to be directly resolved by using corresponding riemann invariants and rankine hugoniot jump conditions so that the crucial and delicate lagrangian treatment in the original grp scheme m ben artzi j falcovitz j comput phys 55 1984 1 32 may be avoided an important difference of resolving the grp of the split 2d rhd equations from the grp of the 1d rhd equations or the non relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the grp of the split 2d rhd equations are nonlinearly coupled through the lorentz factor which is also built in terms of the tangential velocities it is a purely multi dimensional relativistic feature several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2d grp scheme

the generalized riemann problems for compressible fluid flows towards high order the generalized riemann problems grp for nonlinear hyperbolic systems of balance laws in one space dimension are now well known and can be formulated as follows given initial data which are smooth on two sides of a discontinuity determine the time evolution of the solution near the discontinuity in particular the grp of k 1 th order high resolution is based on an analytical evaluation of time derivatives up to kth order which turns out to be dependent only on the spatial derivatives up to kth order while the classical riemann problem serves as a primary building block in the construction of many numerical schemes most notably the godunov scheme the analytic study of grp will lead to an array of grp schemes which extend the godunov scheme currently there are extensive studies on the second order grp scheme which proves to be robust and is capable of resolving complex multidimensional fluid dynamic problems ben artzi and falcovitz 2003 4 in this paper we provide a new approach for solving the grp for the compressible flow system towards high order accuracy the derivation of second order grp solver is more concise compared to those in previous works and the third order grp or quadratic grp is resolved for the first time the latter is shown to be necessary through numerical experiments with strong discontinuities our method relies heavily on the new treatment of rarefaction waves indeed as a main technical step the propagation of singularities argument for the rarefaction fan is simplified by deriving the linear ode systems for the evolution of characteristic derivatives in the x t coordinates for generalized riemann invariants the case of sonic point is incorporated into a general treatment the accuracy of the derived grp solvers is justified and numerical examples are presented for the performance of the resulting schemes

improvement of eigenvalues of sturm liouville problem with t periodic boundary conditions the computation of eigenvalues of sturm liouville problem with t periodic boundary conditions is considered using the richardson extrapolation based on the finite difference the accuracy of the eigenvalues are improved numerical results demonstrate the usefulness of the correction

a direct eulerian grp scheme for compressible fluid flows a direct eulerian generalized riemann problem grp scheme is derived for compressible fluid flows riemann invariants are introduced as the main ingredient to resolve the generalized riemann problem grp directly for the eulerian formulation the crucial auxiliary lagrangian scheme in the original grp scheme is not necessary in the present framework the delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique

accurate computation of higher sturm liouville eigenvalues the computation of eigenvalues of regular sturm liouville problems is considered it is shown that a simple step dependent linear multistep method can be used to reduce the error of the orderk 4 h 2 of the centered finite difference estimate of thek th eigenvalue with uniform step lengthh to an error of orderkh 2 by an appropriate minimization of the local error term of the method one can obtain even more accurate results a comparison of the simple correction techniques of paine de hoog and anderssen and of andrew and paine is given numerical examples demonstrate the usefulness of this correction even for low values ofk

finite element method to solve the spectral problem for arbitrary self adjoint extensions of the laplace beltrami operator on manifolds with a boundary abstract a numerical scheme to compute the spectrum of a large class of self adjoint extensions of the laplace beltrami operator on manifolds with boundary in any dimension is presented the algorithm is based on the characterisation of a large class of self adjoint extensions of laplace beltrami operators in terms of their associated quadratic forms the convergence of the scheme is proved a two dimensional version of the algorithm is implemented effectively and several numerical examples are computed showing that the algorithm treats in a unified way a wide variety of boundary conditions

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

global asymptotic stability for nonlinear difference equations in this paper some sufficient conditions are established for the asymptotic stability of solutions of the nonlinear difference equationx n 1 x n a nx n k b nf x n l n 0 which generalize and improve some exist results in the literature

derivative riemann solvers for systems of conservation laws and ader methods in this paper we first briefly review the semi analytical method e f toro v a titarev solution of the generalized riemann problem for advection reaction equations proc roy soc london 458 2018 2002 271 281 for solving the derivative riemann problem for systems of hyperbolic conservation laws with source terms next we generalize it to hyperbolic systems for which the riemann problem solution is not available as an application example we implement the new derivative riemann solver in the high order finite volume ader advection schemes we provide numerical examples for the compressible euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes

thermodynamical effects and high resolution methods for compressible fluid flows one of the fundamental differences of compressible fluid flows from incompressible fluid flows is the involvement of thermodynamics this difference should be manifested in the design of numerical schemes unfortunately the role of entropy expressing irreversibility is often neglected even though the entropy inequality as a conceptual derivative is verified for some first order schemes in this paper we refine the grp solver to illustrate how the thermodynamical variation is integrated into the design of high resolution methods for compressible fluid flows and demonstrate numerically the importance of thermodynamic effects in the resolution of strong waves as a by product we show that the grp solver works for generic equations of state and is independent of technical arguments

small potential corrections for the discrete eigenvalues of the sturm liouville problem the inverse sturm liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem y qy y holds on 0 fory 0 y 0 and a set of given eigenvalues since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding sturm liouville eigenvalues asymptotic corrections for the s serve to get better estimates forq let k 1 k n be the first eigenvalues of let k be the corresponding discrete eigenvalues obtained by the finite element method for and let k k for the special caseq 0 then starting from an asymptotic correction technique proposed by paine de hoog and anderssen new estimates for the errors bar lambda k lambda k of the corrected discrete eigenvalues bar lambda k lambda k mu k k 2 are obtained and confirm and improve the knowno kh 2 h n 1 behaviour the estimates are based on new sobolev inequalities and on fourier analysis and it is shown that bar lambda k lambda k leqq c 1 kh 2 for 4 c 2 k n 1 2 wherec 1 andc 2 are constants depending onq which tend to 0 for vanishingq

approximations of sturm liouville eigenvalues using differential quadrature dq method the polynomial based differential quadrature pdq and the fourier expansion based differential quadrature fdq methods are applied in this work to compute eigenvalues of the sturm liouville problem it is demonstrated through some examples that accurate results for the first kth eigenvalues of the problem where k 1 2 3 can be obtained by using at least 2k mesh points the results of this work are compared with other published results in the literature

global attractivity of neutral difference equations abstract in this paper we discuss the global attractivity of the neutral difference equation n c n k p n n l 0 n 0 1 2 where is the forward difference operator c 1 1 k l and pn is a sequence of nonnegative real numbers

asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients abstract in this paper we obtain sufficient conditions under which every solution of the neutral differential equation with positive and negative coefficients x t c t x t r p t x t q t x t 0 t t 0 tends to a constant as t

mixed problems for separated variable coefficient wave equations analytic numerical solutions with a priori error bounds this paper deals with the construction of accurate analytic numerical solutions of mixed problems related to the separated variable dependent wave equation utt b t a x uxx 0 0 based on the study of the growth of eigenfunctions of the underlying sturm liouville problems an exact theoretical series solution is firstly obtained explicit bounds allow truncation of the series solution so that the error of the truncated approximation is less than e1 in a bounded domain d x t 0 x l 0 t d since the approximation involves only a finite number of exact eigenvalues i2 1 i n0 the admissible error for the approximated eigenvalues i2 1 i n0 is determined in order to construct an analytical numerical solution of the mixed problem involving only approximated eigenvalues i2 so that the total error is less than e uniformly in d uniqueness of solutions is also treated

self adjoint extensions of the laplace beltrami operator and unitaries at the boundary abstract we construct in this article a class of closed semi bounded quadratic forms on the space of square integrable functions over a smooth riemannian manifold with smooth compact boundary each of these quadratic forms specifies a semi bounded self adjoint extension of the laplace beltrami operator these quadratic forms are based on the lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible the corresponding quadratic forms are semi bounded below and closable finally the representing operators correspond to semi bounded self adjoint extensions of the laplace beltrami operator this family of extensions is compared with results existing in the literature and various examples and applications are discussed

the generalized riemann problem method for the shallow water equations with bottom topography this paper extends the generalized riemann problem method grp to the system of shallow water equations with bottom topography the main contribution is that the generalized riemann problem method j comput phys 1984 55 1 1 32 is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes and at the same step the source term is discretized with an interface method in which only mid point values are plugged in this scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method sgm j comput phys 2001 168 1 1 25 into data reconstruction step and it is also suitable for both steady and unsteady flow simulations we illustrate the accuracy of this scheme by several 1 d and 2 d numerical experiments copyright 2005 john wiley sons ltd

a third order accurate direct eulerian grp scheme for the euler equations in gas dynamics the paper proposes and implements a third order accurate direct eulerian generalized riemann problem grp scheme for one and two dimensional 1d 2d euler equations in gas dynamics it is an extension of the second order accurate grp scheme proposed in ben artzi et al 2006 5 the approximate states in numerical fluxes of the third order accurate grp scheme are derived by using the higher order weno reconstruction of the initial data the limiting values of the time derivatives of the solutions at the singularity point and the jacobian matrix besides the limiting values of the first order time derivatives of fluid variables the second order time derivatives are also needed in developing the present grp scheme and obtained by directly and analytically resolving the local grp in the eulerian formulation via two main ingredients i e the riemann invariants and rankine hugoniot jump conditions unfortunately for the sonic case that the transonic rarefaction wave appears in the grp the jacobian matrix is singular on the sonic line to this end those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation the 2d grp scheme is implemented by using the third order accurate time splitting method several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed grp scheme in comparison to the second order accurate grp scheme

global asymptotic behavior of nonlinear difference equations in this paper the improved 3 2 conditions for the attractivity of the zero solution of nonlinear delay deference equation x n 1 x n f n x g n n 0 1 2 1 1 are obtained some applications are presented to illustrate the advantage of our results

oscillation and nonoscillation of neutral difference equations with positive and negative coefficients abstract in this paper we investigate the oscillation and nonoscillation of the neutral difference equation with variable coefficients x n c n c n r p n x n k q n x n l 0 where pn qn cn n 0 1 2 are real numbers with pn 0 qn 0 cn 0 k l and r are integers with 0 l k 1 r 0 pn qn k l 0 and not identically zero several new sufficient conditions for the oscillation of all solutions of equation are established some of them are sharp our results do not need the usual hypothesis n 0 p n q n k l and improve the all known results in the literature the existence theorems for the positive solutions are also obtained

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

arbitrary high order discontinuous galerkin schemes based on the grp method for compressible euler equations combined with the discontinuous galerkin dg framework the generalized riemann problem grp method is applied to design a grp dg scheme with high order accuracy for compressible euler equations since numerical fluxes with second order accuracy in time are derived by the grp method the reconstruction steps for physical variables in the new scheme are halved compared with the traditional runge kutta discontinuous galerkin rk dg scheme the numerical results are also improved due to more introduced physical information several numerical examples verify the validity of the proposed schemes

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

global existence and mann iterative algorithms of positive solutions for first order nonlinear neutral delay differential equations abstract this paper deals with the first order nonlinear neutral delay differential equation d dt x t p t x t f t x 1 t x 2 t x n t 0 t t 0 where 0 p c t 0 r f c t 0 r n r and l c t 0 r with limt l t for l 1 2 n by using the banach fixed point theorem we prove the global existence of uncountably many bounded positive solutions for the above equation relative to all ranges of the function p construct some mann type iterative algorithms with errors to approximate these positive solutions and discuss several error estimates between the sequences generated by the iterative algorithms and these positive solutions seven examples are presented to illuminate the results obtained in this paper

spectral corrections for sturm liouville problems abstract the numerical solution of the sturm liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction in this paper we use the shooting method both for eigenvalues and eigenfunctions in integrating the corresponding initial value problems we resort to the boundary value method the technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems a technique to estimate the eigenvalue errors is also suggested and seems to be particularly effective for the higher index eigenvalues numerical experiments on some classical sturm liouville problems are presented