multivariate radial basis interpolation for solving quantum fluid dynamical equations this paper proposes a new numerical technique for solving the quantum fluid dynamical equations within the lagrangian description an efficient and accurate numerical scheme is achieved by taking advantage of the smooth field variables obtained via the madelung transformation combined with the radial basis function interpolation applications to the 2d coherent state and a 2d model of no2 photodissociation dynamics show that the present method rivals the split operator method in both efficiency and accuracy the advantage of the new algorithm as a computational tool is expected to prevail for high dimensional systems

generalization of the darboux transformation and generalized harmonic oscillators the darbroux transformation is generalized for time dependent hamiltonian systems which include a term linear in momentum and a time dependent mass the formalism for the n fold application of the transformation is also established and these formalisms are applied for a general quadratic system a generalized harmonic oscillator and a quadratic system with an inverse square interaction up to n 2 among the new features found it is shown for the general quadratic system that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution which is related to the existence of coherent states in the system

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

a galerkin radial basis function method for the schrodinger equation in this article we consider the discretization of the time dependent schrodinger equa tion using radial basis functions rbfs we formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions since we can show that time stability of the discretization is not guaranteed for an rbf collocation method we propose to employ a galerkin ansatz instead for gaussians it is shown that exponential convergence is obtained up to a point where a systematic error from the domain where no basis functions are centered takes over the choice of the shape parameter and of the resolved region is studied numerically compared to the fourier method with periodic boundary conditions the basis functions can be centered in a smaller domain which gives increased accuracy with the same number of points

superconvergence analysis of finite element method for the time dependent schrodinger equation in this paper we consider the two dimensional time dependent schrodinger equation firstly we use the rectangular lagrange type finite element of order p to get a semi discrete scheme of the equation and discuss the superconvergence error estimate in the h 1 norm secondly we use the crank nicolson method in time to get a fully discrete scheme of the equation and the superconvergence estimate in the h 1 norm can be obtained in this scheme finally a numerical example with the order p 1 is provided to verify our theoretical results

a posteriori error control and adaptivity for crank nicolson finite element approximations for the linear schrodinger equation we derive optimal order a posteriori error estimates for fully discrete approximations of linear schrodinger type equations in the l infty l 2 l l 2 norm for the discretization in time we use the crank nicolson method while for the space discretization we use finite element spaces that are allowed to change in time the derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of schrodinger equations the final estimates are obtained using energy techniques and residual type estimators various numerical experiments for the one dimensional linear schrodinger equation in the semiclassical regime verify and complement our theoretical results the numerical implementations are performed with both uniform partitions and adaptivity in time and space for adaptivity we further develop and analyze an existing time space adaptive algorithm to the cases of schrodinger equations the adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem especially for small values of the planck constant

exactly solvable models for the schrodinger equation from generalized darboux transformations exactly solvable models based on darboux transformations of a generalized schrodinger equation are studied the formulation allows not only a unified description of the standard inverse scattering problems at fixed energy and at fixed angular momentum but also includes problems with a linear relationship between lambda 2 and k2 as well as the treatment of lambda 2 or k2 dependent potentials based on a characteristic symmetry property of darboux transformations generalized integral equations are derived and discussed with respect to the standard equations of inverse scattering theory new exactly solvable models for the generalized schrodinger equation are constructed

practical stabilization of a quantum particle in a one dimensional infinite square potential well we consider a nonrelativistic charged particle in a one dimensional infinite square potential well this quantum system is subjected to a control which is a uniform in space time depending electric field it is represented by a complex probability amplitude solution of a schrodinger equation on a one dimensional bounded domain with dirichlet boundary conditions we prove the almost global practical stabilization of the eigenstates by explicit feedback laws

gain in stochastic resonance precise numerics versus linear response theory beyond the two mode approximation in the context of the phenomenon of stochastic resonance sr we study the correlation function the signal to noise ratio snr and the ratio of output over input snr i e the gain which is associated to the nonlinear response of a bistable system driven by time periodic forces and white gaussian noise these quantifiers for sr are evaluated using the techniques of linear response theory lrt beyond the usually employed two mode approximation scheme we analytically demonstrate within such an extended lrt description that the gain can indeed not exceed unity we implement an efficient algorithm based on work by greenside and helfand detailed in the appendix to integrate the driven langevin equation over a wide range of parameter values the predictions of lrt are carefully tested against the results obtained from numerical solutions of the corresponding langevin equation over a wide range of parameter values we further present an accurate procedure to evaluate the distinct contributions of the coherent and incoherent parts of the correlation function to the snr and the gain as a main result we show for subthreshold driving that both the correlation function and the snr can deviate substantially from the predictions of lrt and yet the gain can be either larger or smaller than unity in particular we find that the gain can exceed unity in the strongly nonlinear regime which is characterized by weak noise and very slow multifrequency subthreshold input signals with a small duty cycle this latter result is in agreement with recent analog simulation results by gingl et al icnf 2001 edited by g bosman world scientific singapore 2002 pp 545 548 fluct noise lett 1 l181 2001

numerical analysis for a new non conforming linear finite element on quadrilaterals starting with a short introduction of a new non conforming linear quadrilateral p1 finite element which has been recently proposed by park a study on locking phenomena in finite element methods ph d thesis seoul national university february 2002 and park and sheen p1 nonconforming quadrilateral finite element methods for second order elliptic problems siam j numer anal 41 2 2003 624 640 we examine in detail the numerical behaviour of this element with special emphasis on the treatment of dirichlet boundary conditions efficient matrix assembly and solver aspects furthermore we compare the numerical characteristics of p1 with other low order finite elements also regarding its use for the incompressible navier stokes equations several test examples show the efficiency and reliability of the proposed methods for elliptic second order problems

on the accuracy of the spectral interpolation methods for sampled pseudoperiodic signals abstract the accuracy of a spectral interpolation method is studied on sampled pseudoperiodic signals an attempt is made to compute the accuracy of the characteristics of a particular component in terms of the remaining components contribution the weighting effect is shown to improve the characteristics computation by reducing the remaining components contribution to an order of 1 n 3 where n is the number of the samples the noise effect and the proximity effect are analyzed and the spectral interpolation is compared with a time domain method derived from a kumaresan tufts procedure

locally stabilized p1 nonconforming quadrilateral and hexahedral finite element methods for the stokes equations in this paper we consider locally stabilized pairs p 1 p 1 nonconforming quadrilateral and hexahedral finite element methods for the two and three dimensional stokes equations the stabilization is obtained by adding to the bilinear form the difference between an exact gaussian quadrature rule for quadratic polynomials and an exact gaussian quadrature rule for linear polynomials optimal error estimates are derived in the energy norm and the l 2 norm for the velocity and in the l 2 norm for the pressure in addition numerical experiments to confirm the theoretical results are presented from our numerical results we observe that the proposed stabilized p 1 p 1 nonconforming finite element method shows better performance than the standard method

high resolution finite volume scheme for the quantum hydrodynamic equations the theory of quantum fluid dynamics qfd helps nanotechnology engineers to understand the physical effect of quantum forces although the governing equations of quantum fluid dynamics and classical fluid mechanics have the same form there are two numerical simulation problems must be solved in qfd the first is that the quantum potential term becomes singular and causes a divergence in the numerical simulation when the probability density is very small and close to zero the second is that the unitarity in the time evolution of the quantum wave packet is significant accurate numerical evaluations are critical to the simulations of the flow fields that are generated by various quantum fluid systems a finite volume scheme is developed herein to solve the quantum hydrodynamic equations of motion which significantly improve the accuracy and stability of this method the qfd equation is numerically implemented within the eulerian method a third order modified osher chakravarthy moc upwind centered finite volume scheme was constructed for conservation law to evaluate the convective terms and a second order central finite volume scheme was used to map the quantum potential field an explicit runge kutta method is used to perform the time integration to achieve fast convergence of the proposed scheme in order to meet the numerical result can conform to the physical phenomenon and avoid numerical divergence happening due to extremely low probability density the minimum value setting of probability density must exceed zero and smaller than certain value the optimal value was found in the proposed numerical approach to maintain a converging numerical simulation when the minimum probability density is 10 5 to 10 1 2 the normalization of the wave packet remains close to unity through a long numerical simulation and the deviations from 1 0 is about 10 4 to check the qfd finite difference numerical computations one and two dimensional particle motions were solved for an eckart barrier and a downhill ramp barrier respectively the results were compared to the solution of the schrodinger equation using the same potentials which was obtained using by a finite difference method finally the new approach was applied to simulate a quantum nanojet system and offer more intact theory in quantum computational fluid dynamics

convergence of a semiclassical wavepacket based time splitting for the schrodinger equation we propose a new algorithm for solving the semiclassical time dependent schrodinger equation the algorithm is based on semiclassical wavepackets the focus of the analysis is only on the time discretization convergence is proved to be quadratic in the time step and linear in the semiclassical parameter varepsilon

stability analysis of finite difference schemes for quantum mechanical equations of motion for a pdf involving both space and time variables stability criteria are presently shown to change drastically when the equation contains i as in the quantum mechanical equations of motion it is further noted that the stability of finite difference schemes for quantum mechanical equations of motion depends on both spatial and temporal zoning it is possible to compare a free particle green s function to the solution of a simple diffusion equation and the quantum mechanical motion of a free particle to fresnel diffraction in optics

a stabilized implicit fractional step method for the time dependent navier stokes equations using equal order pairs abstract a stabilized implicit fractional step method for numerical solutions of the time dependent navier stokes equations is presented in this paper the time advancement is decomposed into a sequence of two steps the first step has the structure of the linear elliptic problem the second step can be seen as the generalized stokes problem the two problems satisfy the full homogeneous dirichlet boundary conditions on the velocity on the other hand a locally stabilized term is added in the second step of the schemes it allows one to enhance the numerical stability and efficiency by using the equal order pairs convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method some numerical experiments are also used to demonstrate the efficiency of this new method

mass preserving discontinuous galerkin methods for schrodinger equations we construct analyze and numerically validate a class of mass preserving direct discontinuous galerkin ddg schemes for schrodinger equations subject to both linear and nonlinear potentials up to round off error these schemes preserve the discrete version of the mass of the continuous solution for time discretization we use the crank nicolson for linear schrodinger equations and the strang splitting for nonlinear schrodinger equations so that numerical mass is still preserved at each time step the ddg method when applied to linear schrodinger equations is shown to have the optimal k 1 th order of accuracy for polynomial elements of degree k the numerical tests demonstrate both accuracy and capacity of these methods

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

numerical analysis of the smoluchowski equation using the split operator method abstract we apply the split operator method of feit fleck and steiger to the numerical solution of the smoluchowski equation we study the time evolution of the first few moments and of the equilibrium time correlation function for bistable potentials the time evolution of the probability density obtained by us is compared with the one predicted by the scaling theory of suzuki we also analyze the dependence of the total equilibrium relaxation time on the noise intensity d for small and large values of d the relaxation from an initial condition away from the unstable point is also discussed we find that the one well population is well described by a single exponential law with decay times which grow extremely large as the value of d is decreased

numerical evaluation of coherent state path integrals in quantum dynamics the numerical evaluation of coherent state path integrals for quantum dynamical problems is discussed for one dimensional examples to propagate an initial state we use the normal and antinormal ordered coherent state path integrals combined with a split operator technique dividing the hamiltonian into harmonic and anharmonic parts for numerical purposes integrations must be approximated by quadrature formulae this leads to a matrix multiplication scheme which is systematically tested for the double well and morse potentials the method is accurate for propagation times much longer than the natural time scale of the system and it allows for short as well as long time steps without loss of stability

a finite element formulation for the stokes problem allowing equal velocity pressure interpolation abstract in this paper we study a variational formulation of the stokes problem that accommodates the use of equal velocity pressure finite element interpolations the motivation of this method relies on the analysis of a class of fractional step methods for the navier stokes equations for which it is known that equal interpolations yield good numerical results the reason for this turns out to be the difference between two discrete laplacian operators computed in a different manner the formulation of the stokes problem considered here aims to reproduce this effect from the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity pressure interpolations satisfying a compatibility condition much weaker than the inf sup condition of the standard formulation in particular this condition is fulfilled by the most common equal order interpolations

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

exactly solvable hydrogen like potentials and the factorization method a set of factorization energies is introduced giving rise to a generalization of the schrodinger or infeld and hull factorization for the radial hydrogen like hamiltonian an algebraic intertwining technique involving such factorization energies leads us to derive n parametric families of potentials which in general are almost isospectral to the hydrogen like radial hamiltonians the construction of susy partner hamiltonians with ground state energies greater than the corresponding ground state energy of the initial hamiltonian is also explicitly performed

iterative eigenvalue method using the bloch wave operator formalism with pade approximants and absorbing boundaries we present an iterative method for calculating eigenvalues and eigenvectors of large non hermitian matrices the method uses an iterative procedure to solve the basic bloch equation h ensuremath omega ensuremath omega h ensuremath omega of wave operator theory it involves nonlinear transformations such as the translation of diagonal matrix elements in the complex plane and the use of pad e approximants to treat the strongly coupled states which constitute an intermediate space around the model space in the particular case of floquet eigenstates the further step of adding time dependent absorbing boundaries significantly improves the convergence properties of the iterative calculations

acoustical wave propagator for time domain flexural waves in thin plates in this paper an explicit acoustical wave propagator technique is introduced to describe the time domain evolution of acoustical waves in two dimensional plates a combined scheme with chebyshev polynomial expansion and fast fourier transformation is used to implement the operation of the acoustical wave propagator through this operation the initial wave packet at t 0 is mapped into the wave packet at any instant t 0 by comparison of the results of the exact analytical solution and the euler numerical method we find that this new chebyshev fourier scheme is highly accurate and computationally effective in predicting the acoustical wave propagation in thin plates this method offers an opportunity for future study of dynamic stress concentration and time domain energy flow in coupled structures

structure of positive decompositions of exponential operators the solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data accurate solutions can be systematically derived by decomposing the exponential in a product form for time reversible equations such as the hamilton or the schr odinger equation it is immaterial whether or not the decomposition coefficients are positive in fact most symplectic algorithms for solving classical dynamics contain some negative coefficients for time irreversible systems such as the fokker planck equation or the quantum statistical propagator only positive coefficient decompositions which respect the time irreversibility of the diffusion kernel can yield practical algorithms these positive time steps only forward decompositions are a highly effective class of factorization algorithms this work presents a framework for understanding the structure of these algorithms by a suitable representation of the factorization coefficients we show that specific error terms and order conditions can be solved analytically using this framework we can go beyond the sheng suzuki theorem and derive a lower bound for the error coefficient e vtv by generalizing the framework perturbatively we can further prove that it is not possible to have a sixth order forward algorithm by including only the commutator vtv ensuremath equiv mathbf v t v mathbf the pattern of these higher order forward algorithms is that in going from the 2n mathrm th to the 2n 2 mathrm th order one must include a different commutator v t 2n ensuremath 1 v in the decomposition process

solution of the quantum fluid dynamical equations with radial basis function interpolation the paper proposes a numerical technique within the lagrangian description for propagating the quantum fluid dynamical qfd equations in terms of the madelung field variables r and s which are connected to the wave function via the transformation psi exp r is symbol see text the technique rests on the qfd equations depending only on the form not the magnitude of the probability density rho magnitude of psi 2 and on the structure of r symbol see text 2 ln rho generally being simpler and smoother than rho the spatially smooth functions r and s are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme examples of two dimensional model systems show that the method rivals in both efficiency and accuracy the split operator and chebychev expansion methods the results on a three dimensional model system indicates that the present method is superior to the existing ones especially for its low storage requirement and its uniform accuracy the advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool

solving the bound state schrodinger equation by reproducing kernel interpolation based on reproducing kernel hilbert space theory and radial basis approximation theory a grid method is developed for numerically solving the n dimensional bound state schrodinger equation central to the method is the construction of an appropriate bounded reproducing kernel rk lambda alpha r from the linear operator nabla 2 r lambda 2 where nabla 2 r is the n dimensional laplacian lambda 0 is a parameter related to the binding energy of the system under study and the real number alpha n the proposed sobolev rk lambda alpha r r is shown to be a positive definite radial basis function and it matches the asymptotic solutions of the bound state schrodinger equation numerical tests for the one dimensional 1d morse potential and 2d henon heiles potential reveal that the method can accurately and efficiently yield all the energy levels up to the dissociation limit comparisons are also made with the results based on the distributed gaussian basis method in the 1d case as well as the distributed approximating functional method in both 1d and 2d cases

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

an efficient algorithm based on splitting for the time integration of the schrodinger equation we present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the schrodinger equation when the hamiltonian operator is either time independent or changes slowly with time in the later case the evolution operator can be effectively approximated in a step by step manner first divide the time integration interval in sufficiently short subintervals and then successively solve a schrodinger equation with a different time independent hamiltonian operator in each of these subintervals when discretized in space the schrodinger equation with the time independent hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous hamiltonian system corresponding to a system of coupled harmonic oscillators the particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals sharp local error bounds are obtained for the solution of the linear autonomous hamiltonian system considered in each time subinterval our schemes can be considered in this setting as polynomial approximations to the matrix exponential in a similar way as methods based on chebyshev and taylor polynomials the theoretical analysis supported by numerical experiments performed for different time independent hamiltonians indicates that the new methods are more efficient than schemes based on chebyshev polynomials for all tolerances and time interval lengths the algorithm we present automatically selects for each time subinterval the most efficient splitting scheme among several new optimized splitting methods for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the hamiltonian operator

a stabilized finite element method based on two local gauss integrations for the stokes equations this paper considers a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for the stokes equations approximated by the lowest equal order finite element pairs i e the p 1 p 1 and q 1 q 1 pairs this method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf sup condition optimal error estimates are obtained by applying the standard galerkin technique finally the numerical illustrations agree completely with the theoretical expectations

generalizes darboux transformations classification of inverse scattering methods for the radial schrodinger equation a wide class of darboux transformations of a sturm liouville equation providing a unified treatment of exactly solvable models in quantum scattering theory is considered systematically a classification of darboux transformations is given and the relations between darboux transformations and standard inverse scattering procedures for the radial schrodinger equation including the matrix method of newton sabatier are studied in particular the definitions properties and matrix generalizations of darboux transformations associated with marchenko type integral equations are studied in detail

shape invariance through crum transformation we show in a rigorous way that crum s result regarding the equal eigenvalue spectrum of sturm liouville problems can be obtained iteratively by successive darboux transformations furthermore it can be shown that all neighboring darboux transformed potentials of higher order uk and uk 1 satisfy the condition of shape invariance provided the original potential u does so based on this result we prove that under the condition of shape invariance the nth iteration of the original sturm liouville problem defined solely through the shape invariance is equal to the nth crum transformation

an improved acoustical wave propagator method and its application to a duct structure the pseudospectral time domain method has long been used to describe the acoustical wave propagation however due to the limitation and difficulties of the fast fourier transform fft in dealing with nonperiodic problems the dispersion error is inevitable and the numerical accuracy greatly decreases after the waves arrive at the boundary to resolve this problem the lagrange chebyshev interpolation polynomials were used to replace the previous fft which however brings in an additional restriction on the time step in this paper a mapped chebyshev method is introduced providing the dual benefit of preserving the spectral accuracy and overcoming the time step restriction at the same time three main issues are addressed to assess the proposed technique a spatial derivatives in the system operator and the boundary treatment b parameter selections and c the maximum time step in the temporal operator furthermore a numerical example involving the time domain evolution of wave propagation in

canonical transforms and the efficient integration of quantum mechanical wave equations the integration of time dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry the wave function s energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations we demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude our approach includes the so called kramers henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency

unconditional optimal error estimates for bdf2 fem for a nonlinear schrodinger equation abstract this paper analyzes unconditional optimal error estimates for a 2 step backward differentiation formula bdf2 method for a nonlinear schrodinger equation in the analysis we split an error estimate into two parts one from the spatial discretization and the other from the temporal discretization we present the boundedness of the solution of the time discrete system in the certain strong norms and the error estimates for time discretization by these boundedness and temporal error estimates we obtain the l 2 error estimates without any conditions on a time step size numerical experiments are provided to validate our analysis and check the efficiency of our method

symbolic calculation in development of algorithms split step methods for the gross pitaevskii equation we employ symbolic calculation to perform a systematic study of the accuracy of split step fourier transform methods for the time dependent gross pitaevskii equation gpe provided the most recent approximation for the wave function is always used in the nonlinear atom atom interaction energy every split step algorithm we have tried has the same order time stepping error for the nonlinear gpe and for the linear schroedinger equation

quaternionic madelung transformation and non abelian fluid dynamics abstract we find the quaternionic generalization of the madelung transformation of the schroedinger equation the non abelian nature of the full su 2 gauge group of the quaternionic schroedinger equation leads to a more complex set of equations for the fluid variables we identify the hydrodynamic variables for the quaternionic madelung transformation in order to find equations of motion for these quantities we cast the madelung transformation in the form of a non canonical transformation of the classical hamiltonian field theory for the quaternionic schroedinger equation we first define the canonical bracket and hamiltonian for the quaternionic schroedinger equation and then obtain the non canonical brackets yielding the equations of motion for the quaternionic madelung variables these equations are a particularly natural set of equations for a non abelian fluid and differ from those obtained by bistrovic et al only by a global gauge transformation hep th 0210143 2002 obtaining these equations by a transformation of the quaternionic schroedinger equation makes their simulation by a variety of methods particularly simple

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

some observations on darboux s theorem isospectral hamiltonians and supersymmetry a method to generate isospectral supersymmetric hamiltonians is given which is similar but inequivalent to the usual approach based on darboux s theorem explicit calculations are made for the potential v r 2 sech2 r and the conditionally exactly solvable potential given by v r ar23 b r23 g0 r2

how to calculate the wigner function from phase space we present a method for the direct computation of the wigner function by solving a coupled system of linear partial differential equations in phase space our modified spectral method relies on chebyshev polynomials since this approach allows us to include arbitrary high orders of partial derivatives our procedure is applicable to arbitrary binding potentials we apply our scheme to wigner functions of the harmonic oscillator the morse oscillator and an asymmetric double well potential

radial basis function interpolation in the quantum trajectory method optimization of the multi quadric shape parameter this paper investigates multi quadric radial basis function rbf interpolation and its application in the quantum trajectory method qtm for wave packet propagation in the multi quadric r r2 2 1 2 r is the radial distance from the observation point to the origin of the basis function and is known as the shape parameter due to its affect on the functional form of the basis function the quality of any rbf interpolation scheme is dictated by the choice of this parameter many recent studies have investigated a suitable means for obtaining an optimized time independent parameter the purpose of this study however is to not only to find this optimized shape parameter but also to analyze its time dependence in four different dynamical models the anisotropic harmonic oscillator the downhill ramp the uphill ramp and a harmonic oscillator coupled with a downhill ramp to obtain the optimized shape parameter at each time step an algorithm similar to the leave one out method of cross validation is utilized the results for each of the four models are presented and the feasibility and necessity of employing a shape parameter optimization algorithm for each of the models is discussed

an absolutely stable pressure poisson stabilized finite element method for the stokes equations the pressure poisson stabilized galerkin method for the stokes equation requires the choice of a positive parameter existing theoretical predictions for the range of parameter values that yield stable discretizations seem to be very pessimistic when compared to the computational evidence motivated by this wide gap we first examine a continuous prototype for this class of schemes we show that the prototype is absolutely stable i e it is stable for all parameter values and is optimally accurate we then define a new practical variant of the well known pressure poisson stabilized scheme we prove that the new method is absolutely stable just like its continuous prototype and that it achieves optimal convergence rates with respect to the same mesh independent norms the new method differs from the standard pressure poisson stabilized method in several important aspects first its definition does not degrade to a penalty formulation for the lowest order nodal spaces second the method is absolutely stable with respect to the natural norm for the problem while the standard pressure poisson stabilized method is stable with respect to a mesh dependent norm

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

chebyshev spectral methods for radiative transfer we study the performance of chebyshev spectral methods for time dependent radiative transfer equations starting with a method for one dimensional problems in homogeneous media we show that the modifications needed to consider more general problems such as inhomogeneous media polarization and higher dimensions are straightforward in this method we approximate the spatial dependence of the intensity by an expansion of chebyshev polynomials this yields a coupled system of integro differential equations for the expansion coefficients that depend on angle and time next we approximate the integral operation on the angle variables using a gaussian quadrature rule resulting in a coupled system of differential equations with respect to time using a second order finite difference approximation we discretize the time variable we solve the resultant system of equations with an efficient algorithm that makes chebyshev spectral methods competitive with other methods for radiative transfer equations

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality