numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

a hybrid discontinuous galerkin method for computing the ground state solution of bose einstein condensates a numerical method for computing the ground state solution of bose einstein condensates modeled by the gross pitaevskii equation is presented in this method the three dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions both polynomial and plane wave bases are considered for this purpose and lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution the ground state is computed by an iterative procedure for minimizing the energy the performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher order finite element method

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

some practical experience with the time integration of dissipative equations different methods for the numerical integration of evolution dissipative partial differential equations are tested with the kuramoto sivashinsky equation discretizations in space include galerkin and nonlinear galerkin methods for integration in time three different codes are used including standard stiff ode methods numerical tests show that standard codes for stiff ode render a gain of computing time of several orders of magnitude with respect problem tailored methods

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

efficient spectral computation of the stationary states of rotating bose einstein condensates by preconditioned nonlinear conjugate gradient methods we propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial dis cretization scheme for computing the ground states gs of rotating bose einstein condensates bec modeled by the gross pitaevskii equation gpe we first start by reviewing the classical gradient flow also known as imaginary time imt method which considers the problem from the pde standpoint leading to numerically solve a dissipative equation based on this imt equation we analyze the forward euler fe crank nicolson cn and the classical backward euler be schemes for linear problems and recognize classical power iterations allowing us to derive convergence rates by considering the alternative point of view of minimization problems we propose the preconditioned gradient pg and conjugate gradient pcg methods for the gs computation of the gpe we investigate the choice of the preconditioner which plays a key role in the acceleration of the convergence process the performance of the new algorithms is tested in 1d 2d and 3d we conclude that the pcg method outperforms all the previous methods most particularly for 2d and 3d fast rotating becs while being simple to implement

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

energy minimization related to the nonlinear schrodinger equation in this the window of the sobolev gradient technique to the problem of minimizing a schrodinger functional associated with a nonlinear schrodinger equation we show that gradients act in a suitably chosen sobolev space sobolev gradients can be used in finite difference and finite element settings in a computationally efficient way to find minimum energy states of schrodinger functionals

approximating time evolution related to ginzburg landau functionals via sobolev gradient methods in a finite element setting sobolev gradients have previously 1 been used to approximate time evolution related to a model a functional in a finite difference setting in this journal here a related approach in a finite element setting is discussed

efficient algorithms for the two dimensional ising model with a surface field the bond propagation and site propagation algorithms are extended to the two dimensional 2d ising model with a surface field with these algorithms we can calculate the free energy internal energy specific heat magnetization correlation functions surface magnetization surface susceptibility and surface correlations the method can handle continuous and discrete bond and surface field disorder and is especially efficient in the case of bond or site dilution to test these algorithms we study the wetting transition of the 2d ising model which was solved exactly by abraham we can locate the transition point accurately with a relative error of 10 8 we carry out the calculation of the specific heat and surface susceptibility on lattices with sizes up to 200 2 times 200 the results show that a finite jump develops in the specific heat and surface susceptibility at the transition point as the lattice size increases for lattice size 320 2 times 320 the parallel correlation length exponent is 1 86 while abraham s exact result is 2 0 the perpendicular correlation length exponent for lattice size 160 2 times 160 is 1 05 whereas its exact value is 1 0

stability analysis and continuation for the coupled gross pitaevskii equations abstract we study linear stability analysis for the coupled gross pitaevskii equations cgpes which are used as the governing equations for rotating two component bose einstein condensates bec and ultrarapidly rotating two component rydberg dressed bec the cgpes describe the coexistence of superfluidity and crystallization of bec simultaneously we show numerically that the linear stability of discrete steady state solutions is related to the angular velocity next we describe some continuation algorithms for computing the ground state solutions of the cgpes specifically one of them can efficiently handle the case when long range intra component interactions of the two components are different for ultrarapid rotation our numerical experiments demonstrate a competition between the supersolid crystal structure and the rotating induced vortex lattice which gives rise to new phases

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we describe in detail the application of importance sampling to numerical simulations of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate the samples in monte carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors and it yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency timing and phase fluctuations in a prototypical soliton based communication system

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

shape dependent finite size effect of the critical two dimensional ising model on a triangular lattice using a bond propagation algorithm we study the finite size behavior of the critical two dimensional ising model on a finite triangular lattice with free boundaries in five shapes triangular rhomboid trapezoid hexagonal and rectangular the critical free energy internal energy and specific heat are calculated the accuracy of the free energy reaches 10 26 based on accurate data on several finite systems with linear size up to n 2000 we extract the bulk surface and corner parts of the free energy internal energy and specific heat accurately we confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher order terms in the critical free energy for the rhomboid trapezoid and hexagonal systems which are absent for the triangular and rectangular systems the logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical while the logarithmic edge corrections due to corresponding edges in the specific heat are different the corner internal energy and corner specific heat for angles 3 2 and 2 3 are obtained as well as higher order corrections comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice phys rev e 86 041149 2012 we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal

error analysis and applications of the fourier galerkin runge kutta schemes for high order stiff pdes an integrating factor mixed with runge kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations in this paper the novel hybrid fourier galerkin runge kutta scheme with the aid of an integrating factor is proposed to solve nonlinear high order stiff pdes error analysis and properties of the scheme are provided application to the approximate solution of the nonlinear stiff korteweg de vries the 3rd order pde dispersive equation kuramoto sivashinsky the 4th order pde dissipative equation and kawahara the 5th order pde equations are presented comparisons are made between this proposed scheme and the competing method given by kassam and trefethen it is found that for kdv ks and kawahara equations the proposed method is the best

complete eigenfunctions of linearized integrable equations expanded around a soliton solution complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory in this article we explicitly construct such eigenfunctions for a large class of integrable equations including the kdv nls and mkdv hierarchies we establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions furthermore these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem the key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro differential operator which generates the integrable equation and the linearization operator of the lowest order integrable equation in the same hierarchy we also obtain similar results for the adjoint linearization operator of an integrable equation even though our analysis is conducted only for the kdv nls and mkdv hierarchies similar results are expected for other integrable hierarchies as well we further explicitly present the complete eigenfunctions for the kdv nls and mkdv hierarchy equations and give their inner products thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations

a newton method with adaptive finite elements for solving phase change problems with natural convection we present a new numerical system using finite elements with mesh adaptivity for the simulation of solid liquid phase change systems in the liquid phase the natural convection flow is simulated by solving the incompressible navier stokes equations with boussinesq approximation a variable viscosity model allows the velocity to progressively vanish in the solid phase through an intermediate mushy region the phase change is modeled by introducing an implicit enthalpy source term in the heat equation the final system of equations describing the liquid solid system by a single domain approach is solved using a newton iterative algorithm the space discretization is based on a p2 p1 taylor hood finite elements and mesh adaptivity by metric control is used to accurately track the solid liquid interface or the density inversion interface for water flows the numerical method is validated against classical benchmarks that progressively add strong non linearities in the system of equations natural convection of air natural convection of water melting of a phase change material and water freezing very good agreement with experimental data is obtained for each test case proving the capability of the method to deal with both melting and solidification problems with convection the presented numerical method is easy to implement using freefem software using a syntax close to the mathematical formulation

a composite runge kutta method for the spectral solution of semilinear pdes a new composite runge kutta rk method is proposed for semilinear partial differential equations such as korteweg de vries nonlinear schrodinger kadomtsev petviashvili kp kuramoto sivashinsky ks cahn hilliard and others having high order derivatives in the linear term the method uses fourier collocation and the classical fourth order rk method except for the stiff linear modes which are treated with a linearly implicit rk method the composite rk method is simple to implement indifferent to the distinction between dispersive and dissipative problems and as efficient on test problems for ks and kp as any other generally applicable method

optimizing schrodinger functionals using sobolev gradients applications to quantum mechanics and nonlinear optics in this paper we study the application of the sobolev gradients technique to the problem of minimizing several schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics we show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete fourier basis and a fast fourier transform we show that the sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators

fourier splitting methods for the dynamics of rotating bose einstein condensates abstract we present a new method to propagate rotating bose einstein condensates subject to explicitly time dependent trapping potentials using algebraic techniques we combine magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the hamiltonian that can be computed using only fourier transforms at the cost of solving a small system of polynomial equations the resulting scheme solves the challenging component of the nonlinear hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating bose einstein condensates the method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates e g for small nonlinearities since it retains the near integrable structure of the problem for large nonlinearities the method remains highly efficient if higher order p 2 is sought furthermore we show how it can be adapted to the presence of dissipation terms numerical examples illustrate the performance of the scheme

adaptive multiquadric collocation for boundary layer problems an adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two point boundary value problems using a multiquadric integral formulation the second derivative of the solution is approximated by multiquadric radial basis functions this approach is combined with a coordinate stretching technique the required variable transformation is accomplished by a conformal mapping an iterated sine transformation a new error indicator function accurately captures the regions of the interval with insufficient resolution this indicator is used to adaptively add data centres and collocation points the method resolves extremely thin layers accurately with fairly few basis functions the proposed adaptive scheme is very robust and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally the effectiveness of our new method is demonstrated on two examples with boundary layers and one example featuring an interior layer it is shown in detail how the adaptive method refines the resolution

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

convergence analysis of high order time splitting pseudo spectral methods for rotational gross pitaevskii equations a convergence analysis of time splitting pseudo spectral methods adapted for time dependent gross pitaevskii equations with additional rotation term is given for the time integration high order exponential operator splitting methods are studied and the space discretization relies on the generalized laguerre fourier spectral method with respect to the x y x y variables as well as the hermite spectral method in the z z direction essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations fractional power spaces defined by the principal linear part a sobolev type inequality in a curved rectangle and results on the asymptotical distribution of the nodes and weights associated with gau laguerre quadrature the obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained provided that the problem data satisfy suitable regularity requirements a numerical example confirms the theoretical convergence estimate

numerical algorithms for the direct spectral transform with applications to nonlinear schrodinger type systems we implement two different algorithms for computing numerically the direct zakharov shabat eigenvalue problem on the infinite line the first algorithm replaces the potential in the eigenvalue problem by a piecewise constant approximation which allows one to solve analytically the corresponding ordinary differential equation the resulting algorithm is of second order in the step size the second algorithm uses the fourth order runge kutta method we test and compare the performance of these two algorithms on three exactly solvable potentials we find that even though the runge kutta method is of higher order this extra accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue this limits the usefulness of the runge kutta algorithm to a region inside the unit circle around the origin in the complex plane of the eigenvalues for the computation of the continuous spectrum density this limitation is particularly severe as revealed by the spectral decomposition of thel2 norm of a solution to the nonlinear schrodinger equation we show that no such limitations exist for the piecewise constant algorithm in particular this scheme converges uniformly for both continuous and discrete spectrum components

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

accurate expansions of internal energy and specific heat of critical two dimensional ising model with free boundaries the bond propagation algorithm for the specific heat of the two dimensional ising model is developed and that for the internal energy is completed using these algorithms we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries comparing with previous works phys rev e 86 041149 2012 phys rev e 87 022124 2013 we reach much higher accuracy 10 28 of the internal energy and specific heat compared to the accuracy 10 11 of the internal energy and 10 9 of the specific heat reached in the previous works this leads to much more accurate estimations of the edge and corner terms the exact values of all edge and corner terms are therefore conjectured the accurate forms of finite size scaling for the internal energy and specific heat are determined for the rectangle shaped square lattice with various aspect ratios and various shaped triangular lattice for the rectangle shaped square and triangular lattices and the triangle shaped triangular lattice there is no logarithmic correction terms of order higher than 1 s with s the area of the system for the triangular lattice in rhombus trapezoid and hexagonal shapes there exist logarithmic correction terms of order higher than 1 s for the internal energy and logarithmic correction terms of all orders for the specific heat

sobolev gradients and the ginzburg landau functional we describe a sobolev gradient method for finding minima of the ginzburg landau functional for superconductivity this method leads to a particularly simple algorithm which avoids consideration of the nonlinear boundary conditions associated with the ginzburg landau equations

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

an analysis of schwarz waveform relaxation domain decomposition methods for the imaginary time linear schrodinger and gross pitaevskii equations the aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of classical and quasi optimal schwarz waveform relaxation swr domain decomposition methods applied to the computation of the stationary states of the one dimensional linear and nonlinear schrodinger equations with a space dependent potential although swr methods are currently used for efficiently solving high dimensional partial differential equations their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension except in simple cases in this work we tackle this problem for linear and nonlinear one dimensional schrodinger equations by developing techniques which can be extended to higher dimensional problems and other types of pdes the approach combines the method developed in gander and halpern siam j num anal 45 2 666 697 2007 for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolic asymptotic expansions for computing the stationary states we consider the imaginary time formulation of the schrodinger equation based on the normalized gradient flow ngf method and use a semi implicit euler scheme for the discretization some numerical results in the one dimensional case illustrate the analysis for both the linear schrodinger and gross pitaevskii equations

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

a new sobolev gradient method for direct minimization of the gross pitaevskii energy with rotation in this paper we improve traditional steepest descent methods for the direct minimization of the gross pitaevskii gp energy with rotation at two levels we first define a new inner product to equip the sobolev space h 1 and derive the corresponding gradient second for the treatment of the mass conservation constraint we use a projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods the descent method with these two new ingredients is studied theoretically in a hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the gp energy the new method is implemented in both finite difference and finite element two dimensional settings and is used to compute various complex configurations with vortices of rotating bose einstein condensates the new sobolev gradient method shows better numerical performances compared to classical l 2 or h 1 gradient methods especially when high rotation rates are considered

amplitude energy fluctuations of optical pulses and statistics of the exit distance abstract we consider the propagation of an isolated optical pulse in the presence of additive noise we apply an approach based on the fokker planck equation to determine the probability distribution functions of the propagation distance at which the pulse energy and the pulse amplitude leave a certain region

safe and effective importance sampling abstract we present two improvements on the technique of importance sampling first we show that importance sampling from a mixture of densities using those densities as control variates results in a useful upper bound on the asymptotic variance that bound is a small multiple of the asymptotic variance of importance sampling from the best single component density this allows one to benefit from the great variance reductions obtainable by importance sampling while protecting against the equally great variance increases that might take the practitioner by surprise the second improvement is to show how importance sampling from two or more densities can be used to approach a zero sampling variance even for integrands that take both positive and negative values

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

non monte carlo formulations and computational techniques for the stochastic non linear schrodinger equation stochastic ordinary and partial differential equations sopdes in various forms arise and are successfully utilized in the modeling of a variety of physical and engineered systems such as telecommunication systems electronic circuits cosmological systems financial systems meteorological and climate systems while the theory of stochastic partial and especially ordinary differential equations is more or less well understood there has been much less work on practical formulations and computational approaches to solving these equations in this paper we concentrate on the stochastic non linear schrodinger equation snlse that arises in the analysis of wave propagation phenomena mainly motivated by its predominant role as a modeling tool in the design of optically amplified long distance fiber telecommunication systems we present novel formulations and computational methods for the stochastic characterization of the solution of the snlse our formulations and techniques are not aimed at computing individual realizations i e sample paths for the solution of the snlse a la monte carlo instead starting with the snlse we derive new systems of differential equations and develop associated computational techniques the numerical solutions of these new equations directly produce the ensemble averaged stochastic characterization desired for the solution of the snlse in a non monte carlo manner without having to compute many realizations needed for ensemble averaging

an iterative stochastic method for simulating large deviations and rare events we describe an iterative method capable of determining large deviations responsible for rare events of interest in lightwave systems with additive noise the method makes use of the singular value decomposition svd to efficiently compute the most important directions in state space and a stochastic optimization scheme known as the cross entropy ce method to determine the most probable manner in which these large deviations arise information from the svd and ce steps of the method provides a basis for performing importance sampling with monte carlo simulation allowing one to determine the probabilities of the rare events associated with such large deviations we apply the combined method to investigate some of the mechanisms affecting large amplitude fluctuations in optical systems

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

fourier methods with extended stability intervals for the korteweg de vries equation a full leap frog fourier method for integrating the korteweg de vries kdv equation u t uu x varepsilon u xxx 0 results in an o n 3 stability constraint on the time step where n is the number of fourier modes used this stability limit is much more restrictive than the accuracy limit for many applications in this paper we propose a method for which the stability limit is extended by treating the linear dispersive u xxx term implicitly thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit the implicit method is implemented without solving linear systems by integrating in time in the fourier space and discretizing the nonlinear uu x term by leap frog a second method we propose uses basis functions which solve the linear part of the kdv equation and leap frog for time integration a linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that

a pseudospectral procedure for the solution of nonlinear wave equations with examples from free surface flows an algorithm for the solution of general isotropic nonlinear wave equations is presented the algorithm is based on a symmetric factorization of the linear part of the wave operator followed by its exact integration through an integrating factor in spectral space the remaining nonlinear and forcing terms can be handled with any standard pseudospectral procedure solving the linear part of the wave operator exactly effectively eliminates the stiffness of the original problem characterized by a wide range of temporal scales the algorithm is tested and applied to several problems of three dimensional long surface waves solitary wave propagation interaction diffraction and the generation of waves by flow over slowly varying bottom topography other potential applications include waves in rotating and stratified flows and wave interaction with more pronounced topographic features

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

efficient and spectrally accurate numerical methods for computing ground and first excited states in bose einstein condensates in this paper we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in bose einstein condensates becs we begin with a review on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the gfdn for discretizing the gradient flow we use sine pseudospectral method for spatial derivatives and either backward euler scheme besp or backward forward euler schemes for linear nonlinear terms bfsp for temporal derivatives both besp and bfsp are spectral order accurate for computing the ground and first excited states in bec of course they have their own advantages i for linear case besp is energy diminishing for any time step size where bfsp is energy diminishing under a constraint on the time step size ii at every time step the linear system in bfsp can be solved directly via fast sine transform fst and thus it is extremely efficient and in besp it needs to be solved iteratively via fst by introducing a stabilization term and thus it could be efficient too comparisons between besp and bfsp as well as other existing numerical methods are reported in terms of accuracy and total computational time our numerical results show that both besp and bfsp are much more accurate and efficient than those existing numerical methods in the literature finally our new numerical methods are applied to compute the ground and first excited states in bec in one dimension 1d 2d and 3d with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution

statistics of soliton bearing systems with additive noise we present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise even though the noise is weak we are interested in probabilities of large fluctuations generally non gaussian which are beyond perturbation theory our method is a development of the instanton formalism method of optimal fluctuation based on a saddle point approximation in the path integral we first

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined