error sources in the standard numerical integration of the schrodinger equation an improved method abstract a numerical study of the standard numerov method for the solution of the schrodinger equation is done by looking at the errors of the solution and the limits of precision for this method comparison is made with the results of an improved method that traces back all sources of errors minimizing them and making the error of each of the sources of comparable magnitude to the others in order to optimize the computing time a reduction factor of more than 100 is found for the computing time given a certain accuracy on the other hand fixing a certain computational time a substantial increase in accuracy is gained the improved method proves particularly suitable for the study of weakly bound states potentials with discontinuities singular potentials or those mixing atomic and nuclear degrees of freedom

stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

the sharpness of stability estimates corresponding to a general resolvent condition abstract in the stability analysis of numerical processes the problem arises of establishing moderate upper bounds for the norm of the n th power of given s s matrices recently new upper bounds were derived under generalized versions of the resolvent condition occurring in the kreiss matrix theorem in the present paper the question is answered of whether these new bounds are sharp with respect to their behaviour as n and s

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

a generalization of the numerical range of a matrix abstract a new generalization of the classical numerical range of a matrix is introduced various properties and basic characterizations of this generalized numerical range are established one of these characterizations is used in proving a resolvent inequality for a discretized version of the differential operator d 2 dx 2

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

on the 2d bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approach abstract a series solution for a 2d bioheat pennes conduction model with convective boundary conditions is established by using the classical fourier method to validate and compare results a numerical method for constructing highly accurate numerical solutions is also derived and numerically illustrated both the series solution and the proposed numerical method open interesting possibilities of application in inverse perfusion coefficient identification problems

exponentially fitted numerov methods the error in the estimate of the kth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh length h is o k 6h 4 it is shown that the error can be reduced to o k 3h 4 by using one of the three versions of the exponentially fitted numerov method numerical examples demonstrate the usefulness of this approach even for low values of k

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

several concepts to investigate strongly nonnormal eigenvalue problems eigenvalue analysis plays an important role in understanding physical phenomena however if one deals with strongly nonnormal matrices or operators the eigenvalues alone may not tell the full story a well known tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is varepsilon pseudospectra apart from varepsilon pseudospectra we consider other tools which might help to learn more about the eigenvalue problem viz condition numbers of the eigenvalues condition numbers of sets of eigenvectors and angles between invariant subspaces all these concepts will be studied and compared for both standard and generalized eigenvalue problems the tools can be used to analyze large eigenvalue problems we apply the different concepts to a generalized eigenvalue problem obtained from magnetohydrodynamics in this problem one is interested in an interior part of the spectrum called the alfven spectrum

analytic extrapolation in l norm an alternative approach to qcd sum rules in view of physical applications especially to qcd sum rules the following problem pertaining to analytic extrapolation techniques is studied we are considering amplitudes which are real analytic functions in the complex plane cut along s0 a modelf0 s of the amplitude is given through the values off0 s on some interval s2 s1 withs1 s0 and the values of its discontinuity on these values are approximate and are supplemented by prescribed error channels measured inl norm both on and investigating the compatibility between these data leads to an extremum problem which is solved up to a point where numerical methods can be implemented

stable spectral collocation solutions to a class of benjamin bona mahony initial value problems we are concerned with stable spectral collocation solutions to non periodic benjamin bona mahony bbm modified bbm and benjamin bona mahony burgers bbm b initial value problems on the real axis the spectral collocation is based alternatively on the scaled hermite and sinc functions in order to march in time we use several one step and linear multistep finite difference schemes such that the method of lines mol involved is stable in sense of lax the method based on hermite functions ensures the correct behavior of the solutions at large spatial distances and in long time periods in order to prove the stability we use the pseudospectra of the linearized spatial discretization operators the extent at which the energy integral of bbm model is conserved over time is analyzed for hermite collocation along with various finite difference schemes this analysis has been fairly useful in optimizing the scaling parameter the effectiveness of our approach has been confirmed by some numerical experiments

scheifele two step methods for perturbed oscillators two step methods specially adapted to the numerical integration of perturbed oscillators are obtained the formulation of the methods is based on a refinement of classical taylor expansions due to scheifele g scheifele on the numerical integration of perturbed linear oscillating systems z angew math phys 22 1971 186 210 the key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency w the methods depend on a parameter email protected where h is the stepsize based on the b2 series theory of coleman j p coleman order conditions for a class of two step methods for y f x y ima j numer anal 23 2003 197 220 we derive the order conditions of this new type of method the linear stability and phase properties are examined the theory is illustrated with some fourth and fifth order explicit schemes numerical results carried out on an assortment of test problems such as the integration of the orbital motion of earth satellites show the relevance of the theory

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

a symmetric matrix method for schrodinger eigenstates it is shown how the one dimensional schrodinger eigenvalue equation can be transformed into a symmetric generalised matrix eigenproblem with a local truncation error of only delta 4 where delta is the step size as in the numerov algorithm standard sparse matrix library packages are available for the solution and it is demonstrated that groups of eigenvalues and their associated eigenvectors wavefunctions can be determined simultaneously with high precision and speed

on the use of stability regions in the numerical analysis of initial value problems this paper deals with the stability analysis of one step methods in the numerical solution of initial boundary value problems for linear ordinary and partial differential equations restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size these restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration the errors in the one step methods are measured in arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of the heat equation and some other differential equations where the error growth is measured in the maximum norm

the equivalence of l2 stability the resolvent condition and strict h stability abstract the kreiss matrix theorem asserts that a family of n n matrices is l 2 stable if and only if either a resolvent condition r or a hermitian norm condition h is satisfied we give a direct considerably shorter proof of the power boundedness of an n n matrix satisfying r sharpening former results by showing that power boundedness depends at most linearly on the dimension n we also show that l 2 stability is characterized by an h condition employing a general h numerical radius instead of the usual h norm thus generalizing a sufficient stability criterion due to lax and wendroff

symmetric matrix methods for schrodinger eigenvectors the one dimensional single particle schrodinger eigenvalue equation is represented by a generalized matrix eigenvalue equation of the form a psi ea psi where a is a symmetric matrix a is a symmetric positive definite matrix and psi is a column vector the local error in making the representation is proportional to delta 4 where delta is the step size several methods of solution which make use of the sparsity of the matrices are investigated

finite difference scheme to solve schrodinger equations comparisons are made among several three node finite difference schemes fds s for solving time independent schr odinger equations it is shown that the mickens fds is although exact in some special cases generally two orders lower than the numerov fds an alternative fds the combined numerov mickens fds is introduced it has advantages of both the numerov and the mickens fds s numerical comparison among these fds s is presented

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

construction of asymptotic solutions to discrete bessel equations abstract we construct four finite difference models for the bessel differential equation they correspond respectively to the standard numerov mickens ramadhani and combined numerov mickens schemes the asymptotic behavior of the solutions to these difference equations is calculated and compared to the asymptotic solution of the bessel differential equation these results are then related to the problem of numerically integrating schrodinger type ordinary differential equations

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

on a generalization of the resolvent condition in the kreiss matrix theorem this paper deals with a condition on the resolvent of s x s matrices a in one of the equivalent assertions of the kreiss matrix theorem the spectral norm of the resolvent of a at 4 must satisfy an inequality for all 4 lying outside the unit disk in c we consider a generalization in which domains different from the unit disk and more general norms are allowed under this generalized resolvent condition an upper bound is derived for the norms of the nth powers of s x s matrices b here b depends on a via a relation b 0 a where p is an arbitrary rational function the upper bound grows linearly with s 1 and is independent of n 1 this generalizes an upper bound occurring in the kreiss theorem where b a like the classical kreiss theorem the upper bound derived in this paper can be used in the stability analysis of numerical methods for solving differential equations

about the sharpness of the stability estimates in the kreiss matrix theorem one of the conditions in the kreiss matrix theorem involves the resolvent of the matrices a under consideration this so called resolvent condition is known to imply for all n 1 the upper bounds an ek n 1 and an ek n 1 here is the spectral norm k is the constant occurring in the resolvent condition and the order of a is equal to n 1 1 it is a long standing problem whether these upper bounds can be sharpened for all fixed k 1 to bounds in which the right hand members grow much slower than linearly with n 1 and with n 1 respectively in this paper it is shown that such a sharpening is impossible the following result is proved for each e 0 there are fixed values c 0 k 1 and a sequence of n 1 n 1 matrices an satisfyirig the resolvent condition such that an n c n 1 1 e c n 1 1 e for n n 1 2 3 the result proved in this paper is also relevant to matrices a whose e pseudospectra lie at a distance not exceeding ke from the unit disk for all e 0

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

solution of an extremum problem pertaining to analytic extrapolation techniques a preliminary problem appearing in analytic extrapolation procedures when the physical data are given on a line contained in the analyticity domain and their errors are measured through the norm of the supremum is solved as an interesting by product blaschke functions are constructed which in some sense generalize the usual chebyshev polynomials

resolvent conditions and bounds on the powers of matrices with relevance to numerical stability of initial value problems we deal with the problem of establishing upper bounds for the norm of the nth power of square matrices this problem is of central importance in the stability analysis of numerical methods for solving linear initial value problems for ordinary partial or delay differential equations a review is presented of upper bounds which were obtained in the literature under the resolvent condition occurring in the kreiss matrix theorem as well as under variants of that condition moreover we prove new bounds under resolvent conditions which generalize some of the reviewed ones the paper concludes by applying one of the new upper bounds in a stability analysis of the trapezoidal rule for delay differential equations

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

bounds for the continuation of perturbative results to the spectral region the problem of analytic continuation to the boundary of the holomorphy domain from both continuous and discrete interior sets has recently been the subject of detailed analyses this problem is important in phenomenological applications but is also of interest in theoretical calculations e g in attempting to evaluate the parameters of resonances or other nonperturbative effects in qcd because of the inherent instability of the continuation problem it is necessary to introduce additional criteria which should be physically based to select the right continuation function in this paper the results thus obtained for continuation from a continuum are examined for stability and bounds are derived for the errors on the boundary in terms of the uncertainty of the input data the procedure is shown to be stable in the sense that these bounds tend to zero as the data errors go to zero

phase fitted and amplification fitted two step hybrid methods for y f x y this paper provides a theoretical framework for a new type of phase fitted and amplification fitted two step hybrid ftsh methods which is introduced by the author in h van de vyver a phase fitted and amplification fitted explicit two step hybrid method for second order periodic initial value problems internat j modern phys c 17 2006 663 675 the methods constitute a modification of dissipative two step hybrid methods in the sense that two free parameters are added to eliminate the phase lag and the amplification error the methods are useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize the algebraic order zero stability stability and phase properties are examined the theory is illustrated with sixth order explicit ftsh methods numerical results carried out on an assortment of test problems show the relevance of the theory

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

chebyshev pseudospectral method for computing numerical solution of convection diffusion equation abstract a method for computing highly accurate numerical solutions of 1d convection diffusion equations is proposed in this method the equation is first discretized with respect to the spatial variable transforming the original problem into a set of ordinary differential equations and then the resulting system is integrated in time by the fourth order runge kutta method spatial discretization is done by using the chebyshev pseudospectral collocation method before describing the method we review a finite difference based method by salkuyeh d khojasteh salkuyeh on the finite difference approximation to the convection diffusion equation appl math comput 179 2006 79 86 and contrary to the proposal of the author we show that this method is not suitable for problems involving time dependent boundary conditions which calls for revision stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out superiority of the proposed method over a revised version of salkuyeh s method is verified by numerical examples

stability of a chebychev pseudospectral solution of the wave equation with absorbing boundaries abstract stability of the pseudospectral chebychev collocation solution of the two dimensional acoustic wave problem with absorbing boundary conditions is investigated the continuous one dimensional problem with one absorbing boundary and one dirichlet boundary has previously been shown to be far from normal consequently the spectrum of that problem says little about the stability behavior of the solution our analysis proves that the discrete formulation with dirichlet boundaries at all boundaries is near normal and hence the formulation with absorbing boundaries at all boundaries either for one dimensional or two dimensional wave propagation is not far from normal the near normality follows from the near normality of the second order derivative pseudospectral differential operator further the nearness to normality is independent of the boundary discretization stability limits on the timestep are however dependent on the boundary operator with an explicit euler method having the most restrictive condition the crank nicolson implementation has a stability limit the same as the dirichlet formulation furthermore in this case the restriction scales by 1 2 in moving from one dimension to two dimensions exactly as in the central finite difference approximation numerical results confirm the predicted values on allowable timesteps obtained from a spectral analysis for both chebychev and modified chebychev implementations we conclude that the spectrum of the evolution operator is informative for predicting the behavior of the numerical solution

asymptotic properties of solutions to discrete coulomb equations abstract we construct two finite difference models for the coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles these difference equations correspond to the standard and mickens ramadhani schemes for the coulomb equation our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the coulomb differential equation in particular the form of the anomalous phase term is examined

stability estimates for families of matrices of nonuniformly bounded order this paper addresses the problem of establishing upper bounds for the norm of the nth power of square matrices the focus is on upper bounds that are valid for families of matrices of finite but nonuniformly bounded order such upper bounds are relevant to the stability analysis of numerical methods for solving differential equations in the famous kreiss matrix theorem a condition on the resolvent of matrices occurs which implies that the norm of the nth power does not grow faster than linearly with n in this paper a slightly stronger version of this resolvent condition is studied which is often satisfied in cases of practical interest we prove that this version implies growth at a rate that is essentially lower than in the case of the classical kreiss resolvent condition we consider also a condition on the iterated resolvent of matrices which is related to one of the assumptions occurring in the hille yosida theorem this condition is known to imply growth at a rate b n we prove that when a slightly stronger version of the iterated resolvent condition is in force there is growth at a rate that is at most n 1 1 with 0 q p i

pseudospectra of waveform relaxation operators abstract the performance of the waveform relaxation method for solving systems of odes depends largely on the choices that are made for splitting size of time window and preconditioning although it is known that superlinear convergence is obtained on finite time windows the convergence may be slow in the first few iterations we propose the use of pseudospectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of odes through pseudospectral radii we can examine the effect of preconditioning and overlapping on the rate of convergence we may also use this to estimate a suitable size of the time window numerical experiments performed on a system of odes arising from the discretization of an advection diffusion equation confirm the validity of the obtained estimates

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

on the uniform power boundedness of a family of matrices and the applications to one leg and linear multistep methods we study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy dt y and constant stepsizeh lets be the region of the absolute stability of the method and letd be a closed subset ofs on the riemann sphere mathbb c it is shown that the solutions of these difference equations are bounded forn 0 uniformly for h d s is itself closed in mathbb c iff s is free of cusps the question is studed by means of contractivity analysis and a matrix theorem derived from the matrix theorem of kreiss

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