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On s'interesse a une approximation par elements finis lineaire non-conforme d'un fluide non-Newtonien, dont la viscosite suit une loi de Carreau pseudo-plastique. Un encadrement optimal de l'erreur, a la fois pour la vitesse et la pression, est determine a priori. Par ailleurs sont presentes des estimateurs d'erreur a posteriori, bases sur une evaluation locale de l'erreur residuelle. Ces estimateurs permettent une majoration globale et une minoration locale de l'erreur. Enfin, des applications numeriques viennent valider l'encadrement a priori de l'erreur.

We study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.

We study analytically and asymptotically as well as numerically ground states and dynamics of two-component spin-orbit-coupled Bose-Einstein condensates (BECs) modeled by the coupled Gross-Pitaevskii equations (CGPEs). In fact, due to the appearance of the spin-orbit (SO) coupling in the two-component BEC with a Raman coupling, the ground state structures and dynamical properties become very rich and complicated. For the ground states, we establish the existence and non-existence results under different parameter regimes, and obtain their limiting behaviors and/or structures with different combinations of the SO and Raman coupling strengths. For the dynamics, we show that the motion of the center-of-mass is either non-periodic or with different frequency to the trapping frequency when the external trapping potential is taken as harmonic and the initial data is chosen as a stationary state (e.g. ground state) with a shift, which is completely different from the case of a two-component BEC without the SO coupling, and obtain the semiclassical limit of the CGPEs in the linear case via the Wigner transform method. Efficient and accurate numerical methods are proposed for computing the ground states and dynamics, especially for the case of box potentials. Numerical results are reported to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomenon in the SO-coupled BECs., Comment: 27 pages, 4 figures

We propose a sharp interface model for simulating solid-state dewetting where the surface energy is (weakly) anisotropic. The morphology evolution of thin films is governed by surface diffusion and contact line migration. The mathematical model is based on an energy variational approach. Anisotropic surface energies lead to multiple solutions of the contact angle equation at contact points. Introduction of a finite contact point mobility is both physically based and leads to robust, unambiguous determination of the contact angles. We implement the mathematical model in an explicit finite difference scheme with cubic spline interpolation for evolving marker points. Following validation of the mathematical and numerical approaches, we simulate the evolution of thin film islands, semi-infinite films, and films with holes as a function of film dimensions, Young's angle $\theta_i$, anisotropy strength and crystal symmetry, and film crystal orientation relative to the substrate normal. We find that the contact point retraction rate can be well described by a power-law, $l \sim t^n$. Our results demonstrate that the exponent $n$ is not universal -- it is sensitive to the Young's angle $\theta_i$ (and insensitive to anisotropy). In addition to classical wetting (where holes in a film heal) and dewetting (where holes in a film grow), we observe cases where a hole through the film heals but leave a finite size hole/bubble between the continuous film and substrate or where the hole heals leaving a continuous film that is not bonded to the substrate. Surface energy anisotropy (i) increases the instability that leads to island break-up into multiple islands, (ii) enhances hole healing, and (iii) leads to finite island size even under some conditions where the Young's angle $\theta_i$ suggests that the film wets the substrate., Comment: 25 pages, 13 figures

In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in \({\frak R}^2\). A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective.

In this paper the numerical simulation of the steady incompressible viscous flow in a no-slip channel is considered. A sequence of approximate nonlocal artificial boundary conditions on a given segment artificial boundary is derived by a system of linearized Navier?Stokes equations and spectral techniques. Then the original problem is reduced to a boundary value problem in a bounded computational domain. The numerical examples show that these artificial boundary conditions are very effective and are also more accurate than Dirichlet and Neumann boundary conditions, which are often used in the engineering literature.

We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\bx)$ and a density function $\rho(\bx)=|\psi(\bx)|^2$, for some complex-valued wave function $\psi(\bx)$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\widehat{U}(\bk)$ has a singularity at the origin $\bk={\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\bk$ which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in $\hat{U}(\bk)$ at the origin is canceled, so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function $\rho(\bx)$ is smooth and decays sufficiently fast as $\bx \rightarrow \infty$. More precisely, the calculation requires $O(N\log N)$ operations, where $N$ is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm., Comment: 17 pages, 0 figure

In this paper, we begin with the nonlinear Schrodinger/Gross-Pitaevskii equation (NLSE/GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported., 23 pages 1 figure

We propose an exponential wave integrator sine pseudospectral (EWI-SP) method for the nonlinear Schr\"{o}dinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. The NLSW is NLS perturbed by the wave operator with strength described by a dimensionless parameter $\varepsilon\in(0,1]$. As $\varepsilon\to0^+$, the NLSW converges to the NLS and for the small perturbation, i.e. $0, Comment: 23 pages

We derive rigorously one- and two-dimensional mean-field equations for cigar- and pancake-shaped Bose-Einstein condensates (BEC) with higher order interactions (HOI). We show how the higher order interaction modifies the contact interaction of the strongly confined particles. Surprisingly, we find that the usual Gaussian profile assumption for the strongly confining direction is inappropriate for the cigar-shaped BEC case, and a Thomas-Fermi type profile should be adopted instead. Based on the derived mean field equations, the Thomas-Fermi densities are analyzed in presence of the contact interaction and HOI. For both box and harmonic traps in one, two and three dimensions, we identify the analytical Thomas-Fermi densities, which depend on the competition between the contact interaction and the HOI., Comment: 7 figures

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature., 135 pages and 10 figures

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $00$ as step size, which imply that the two MTIs converge uniformly with linear convergence rate at $O(\tau)$ for $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0, Comment: 39 pages and 1 figure

We study numerically the long-time dynamics of a one-dimensional Bose-Einstein condensate expanding in a speckle or impurity disorder potential. Using the mean-field Gross-Pitaevskii equation, we demonstrate subdiffusive spreading of the condensate for long times. We find that interaction-assisted hopping between normal modes leads to this subdiffusion. A possible (partial) reason why the root-mean-square (rms) width saturates in the experiment [Nature (London) 453, 891 (2008)] is provided. We suggest that observing both the participation length and the rms width of a condensate, rather than only the rms width, could provide a more complete description of the long-time behavior of ultracold atoms in disorder potentials. Our study confirms subdiffusive spreading in spatially continuous disordered interacting models and highlights new features which spatially discrete models do not possess.

We propose a framework for calculating scattering and bound state properties in anisotropic two-dimensional potentials. Using our method, we derive systematic approximations of partial wave phase shifts and binding energies. Moreover, the method is suitable for efficient numerical computations. We calculate the s-wave phase shift and binding energy of polar molecules in two layers polarized by an external field along an arbitrary direction. We find that scattering depends strongly on their polarization direction and that absolute interlayer binding energies are larger than thermal energies at typical ultracold temperatures., 5 pages, 2 figures

We propose a two-dimensional model for a multilayer stack of dipolar Bose-Einstein condensates formed by a strong optical lattice. We derive effective intra- and interlayer dipole-dipole interaction potentials and provide simple analytical approximations for a given number of lattice sites at arbitrary polarization. We find that the interlayer dipole-dipole interaction changes the transverse aspect ratio of the ground state in the central layers depending on its polarization and the number of lattice sites. The changing aspect ratio should be observable in time of flight images. Furthermore, we show that the interlayer dipole-dipole interaction reduces the excitation energy of local perturbations, affecting the development of a roton minimum., Comment: 10 pages, 6 figures. Submitted 2011

We design a discrete artificial boundary condition for the steady incompressible Navier-Stokes equations in stream function vorticity formulation in an infinite channel. The new boundary condition is derived from a linearized Navier-Stokes system and a fast iterative method. Numerical experiments for the nonlinear Navier-Stokes equations are presented. The discrete artificial boundary condition is compared to Dirichlet and Neumann boundary conditions for the flow over a forward or backward facing step and for flow past a rectangular cylinder. Numerical results show that our boundary condition is very effective.

Dislocation Dynamics in 2 + e Dimensions: Slip Planes, Thin Films, and Grain Boundaries (Y Xiang et al.) Introduction to Molecular Dynamics Simulations (X-T Li) Modeling and Numerical Approximation of Two-Phase Incompressible Flows by a Phase-Field Approach (J Shen) Introduction to Numerical Stochastic Homogenization and the Related Computational Challenges: Some Recent Developments (A Anantharaman et al.).

We derive rigorous one- and two-dimensional mean-field equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equations and the analytical density profiles agree well with numerical solutions of the full Gross-Pitaevskii equation while being more efficient to compute., 11 pages, 5 figures. Updated references

New efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a dipolar interaction potential. Due to the high singularity in the dipolar interaction potential, it brings significant difficulties in mathematical analysis and numerical simulations of dipolar BECs. In this paper, by decoupling the two-body dipolar interaction potential into short-range (or local) and long-range interactions (or repulsive and attractive interactions), the GPE for dipolar BECs is reformulated as a Gross-Pitaevskii-Poisson type system. Based on this new mathematical formulation, we prove rigorously existence and uniqueness as well as nonexistence of the ground states, and discuss the existence of global weak solution and finite time blow-up of the dynamics in different parameter regimes of dipolar BECs. In addition, a backward Euler sine pseudospectral method is presented for computing the ground states and a time-splitting sine pseudospectral method is proposed for computing the dynamics of dipolar BECs. Due to the adoption of new mathematical formulation, our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem. Extensive numerical examples in 3D are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar BECs.

We propose efficient and accurate numerical methods for computing the ground-state solution of spin-1 Bose-Einstein condensates subjected to a uniform magnetic field. The key idea in designing the numerical method is based on the normalized gradient flow with the introduction of a third normalization condition, together with two physical constraints on the conservation of total mass and conservation of total magnetization. Different treatments of the Zeeman energy terms are found to yield different numerical accuracies and stabilities. Numerical comparison between different numerical schemes is made, and the best scheme is identified. The numerical scheme is then applied to compute the condensate ground state in a harmonic plus optical lattice potential, and the effect of the periodic potential, in particular to the relative population of each hyperfine component, is investigated through comparison to the condensate ground state in a pure harmonic trap.

Since its realization in dilute bosonic atomic gases [7], [23], Bose-Einstein condensation of alkali atoms and hydrogen has been produced and studied extensively in the laboratory [1], and has permitted an intriguing glimpse into the macroscopic quantum world. In view of potential applications [38], [61], [63], the study of quantized vortices, which are well-known signatures of superfluidity, is one of the key issues. In fact, bulk superfluids are distinguished from normal fluids by their ability to support dissipationless flow. Such persistent currents are intimately related to the existence of quantized vortices, which are localized phase singularities with integer topological charge [39]. The superfluid vortex is an example of a topological defect that is well known in superconductors [52] and in liquid helium [33]. The occurrence of quantized vortices in superfluids has been the focus of fundamental theoretical and experimental work [33]. Different research groups have obtained quantized vortices in Bose-Einstein condensates (BECs) experimentally, e.g., the JILA group [35], [57], the ENS group [56] and the MIT group [1], [32]. Currently, there are at least two typical ways to generate quantized vortices from a BEC ground state: (i) impose a laser beam rotating with an angular velocity on the magnetic trap holding the atoms to create a harmonic anisotropic potential [51], [4], [75]; or (ii) add to the stationary magnetic trap a narrow, moving Gaussian potential, representing a far-blue detuned laser [49], [50], [24], [25], [10], [12].

In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose-Einstein condensates (BEC) by using the normalized gradient flow or imaginary time method. The key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization. This allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of single component BEC to compute the ground state of spin-1 BEC. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method (BFSP), is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BEC with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method., Comment: 25 pages, 12 figures

A new continuous configuration time-dependent self-consistent field method has been developed to study polyatomic dynamical problems by using the discrete variable representation for the reaction system, and applied to a reaction system coupled to a bath. The method is very efficient because the equations involved are as simple as those in the traditional single configuration approach, and can account for the correlations between the reaction system and bath modes rather well.

We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numer- ically. We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic conflnement and present its semiclassical scaling and geometrical optics. We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and flnd that symmetric and central vortex states are independent of the angular rotational momentum. We perform numerical experiments computing these states using a continuous normal- ized gradient ∞ow (CNGF) method with a backward Euler flnite difierence (BEFD) discretization. Ground, symmetric and central vortex states, as well as their energy conflgurations, are reported in 2D and 3D for a rotating BEC. Through our numerical study, we flnd various conflgurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of difierent states in a strong replusive interaction regime. Finally we report the critical angular velocity at which the ground state loses symmetry, numerical veriflcation of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.

We numerically study the behavior of collapsing and exploding condensates using the parameters of the experiments by E.A. Donley et al. [Nature, 412, 295, (2001)]. Our studies are based on a full three-dimensional numerical solution of the Gross-Pitaevskii equation (GPE) including three body loss. We determine the three body loss rate from the number of remnant condensate atoms and collapse times and obtain only one possible value which fits with the experimental results. We then study the formation of jet atoms by interrupting the collapse and find very good agreement with the experiment. Furthermore we investigate the sensitivity of the jets to the initial conditions. According to our analysis the dynamics of the burst atoms is not described by the GPE with three body loss incorporated., Comment: 9 pages, 10 figures

We review some modelling and numerical aspects in quantum kinetic theory for a gas of interacting bosons and we try to explain what makes Bose-Einstein condensation in a dilute gas mathematically interesting and numerically challenging. Particular care is devoted to the development of efficient numerical schemes for the quantum Boltzmann equation that preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop numerical methods that are able to capture the challenging phenomenon of bosons condensation. We also show that the resulting schemes can be evaluated with the use of fast algorithms. In order to study the evolution of the condensate wave function the Gross-Pitaevskii equation is presented together with some schemes for its efficient numerical solution.

In this paper, we prove the energy diminishing of a normalized gradient flow which provides a mathematical justification of the imaginary time method used in physical literatures to compute the ground state solution of Bose-Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the normalized gradient flow. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD), the other one is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for 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 energy diminishing property of the normalized gradient flow. Numerical results in 1d, 2d and 3d 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 normalized gradient flow can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function., 28 pages, 6 figures

In this paper, we propose a new numerical method to compute the ground state solution of trapped interacting Bose-Einstein condensation (BEC) at zero or very low temperature by directly minimizing the energy functional via finite element approximation. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and show how to reduce it to 2d and 1d GPEs. The ground state solution is formulated by minimizing the energy functional under a constraint, which is discretized by the finite element method. The finite element approximation for 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry are presented in detail and approximate ground state solutions, which are used as initial guess in our practical numerical computation of the minimization problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. Furthermore, comparisons between the ground state solutions and their Thomas-Fermi approximations are also reported. Extension of the numerical method to compute the excited states of GPE is also presented., Comment: 33 pages, 22 figures

In this note we review the time-splitting spectral method, recently studied by the authors, for linear[2] and nonlinear[3] Schrodinger equations (NLS) in the semiclassical regimes, where the Planck constant ɛ is small. The time-splitting spectral method under study is unconditionally stable and conserves the position density. Moreover it is gauge invariant and time reversible when the corresponding Schrodinger equation is. Numerical tests are presented for linear, for weak/strong focusing/defocusing nonlinearities, for the Gross-Pitaevskii equation and for current-relaxed quantum hydrodynamics. The tests are geared towards understanding admissible meshing strategies for obtaining ‘correct’ physical observables in the semi-classical regimes. Furthermore, comparisons between the solutions of the nonlinear Schrodinger equation and its hydrodynamic semiclassical limit are presented.

In this note we review the random projection method, recently introduced by the authors, for numerical simulations of the hyperbolic conservation laws with stiff reaction terms: $$ {U_t} + F{(U)_x} = \frac{1}{\varepsilon }\Psi (U) $$ . In this problem, the reaction timeEis small, making the problem numerically stiff. A classic spurious numerical phenomenon - the incorrect shock speed - occurs when the reaction time scale is not properly resolved numerically. The random projection method, a fractional step method that solves the homogeneous convection by any shock capturing method, followed by a random projection for the reaction term, was introduced in [1] to handle this numerical difficulty. For a scalar model problem, one can prove that the random projection methods capture the correct shock speed with a first order accuracy, if a monotonicity-preserving method is used in the convection step. This method can be extended to compute stiff multi-species detonation waves by randomizing the ignition temperatures. Numerical results of a problem with five species and two reactions will be presented.

Motivated by a recent experiment [J. Catani et al., arXiv:1106.0828v1 preprint, 2011], we study breathing oscillations in the width of a harmonically trapped impurity interacting with a separately trapped Bose gas. We provide an intuitive physical picture of such dynamics at zero temperature, using a time-dependent variational approach. In the Gross-Pitaevskii regime we obtain breathing oscillations whose amplitudes are suppressed by self trapping, due to interactions with the Bose gas. Introducing phonons in the Bose gas leads to the damping of breathing oscillations and non-Markovian dynamics of the width of the impurity, the degree of which can be engineered through controllable parameters. Our results reproduce the main features of the impurity dynamics observed by Catani et al. despite experimental thermal effects, and are supported by simulations of the system in the Gross-Pitaevskii regime. Moreover, we predict novel effects at lower temperatures due to self-trapping and the inhomogeneity of the trapped Bose gas., 7 pages, 3 figures

We study the interaction-induced localization -- the so-called self-trapping -- of a neutral impurity atom immersed in a homogeneous Bose-Einstein condensate (BEC). Based on a Hartree description of the BEC we show that -- unlike repulsive impurities -- attractive impurities have a singular ground state in 3d and shrink to a point-like state in 2d as the coupling approaches a critical value. Moreover, we find that the density of the BEC increases markedly in the vicinity of attractive impurities in 1d and 2d, which strongly enhances inelastic collisions between atoms in the BEC. These collisions result in a loss of BEC atoms and possibly of the localized impurity itself., Comment: 7 pages, 5 figures

In this paper we consider numerical simulation of incompressible viscous flow around an obstacle in velocity-pressure formulation. Two horizontal straight line artificial boundaries are introduced and the original flow is approximated by a flow in an infinite channel with slip boundary condition on the wall. Then two vertical segment artificial boundaries are introduced to limit the channel to a bounded computational domain. In the region of the channel between the vertical boundaries and infinity, the velocity of the flow is almost a constant vector, in which the Navier-Stokes equations can be linearised by Oseen equations and thus a general solution can be derived by using separation of variables. Artificial boundary conditions on the vertical segments are then designed by imposing the continuity of velocity and normal stress. Therefore, the original problem is reduced to a boundary value problem on a bounded computational domain. Numerical example shows that our artificial boundary conditions are very effective.

We design an artificial boundary condition for the steady incompressible Navier-Stokes equations in streamfunction-vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived from the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier-Stokes equations is presented. The artificial boundary condition is compared with Dirichlet and Neumann boundary conditions for the flow past a rectangular cylinder in a flat channel.

This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schr��dinger equations (NLS). The method is explicit, unconditionally stable and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing nonlinear Schr��dinger equations in 2d with a linear, cubic or a quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter $\dt$ is larger than a threshold value $\dt_{\rm th}$. We note that our method can also be applied to solve the 3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC)., SIAM Journal on Numerical Analysis, to appear

In this paper we present error estimates for the finite element approximation of linear elliptic problems in unbounded domains that are outside an obstacle and a semi-infinite strip in the plane. The finite element approximation is formulated on a bounded domain using a nonlocal approximate artificial boundary condition. In fact there is a family of approximate boundary conditions with increasing accuracy (and computational cost) for a given artificial boundary. Our error estimates are based on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. Numerical examples for Poisson's problem outside a circle and in a semi-infinite strip are presented. Numerical results demonstrate the performance of our error estimates.

In this paper we consider the numerical simulation for the problem of infinite elastic foundation. A half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines. Then the original problem is reduced to a boundary value problem on a bounded computational domain. Furthermore, finite element approximation of this boundary value problem defined in the bounded domain is considered. The numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the Neumann boundary condition which is often used in engineering literatures.