ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 05 Dec 2023 21:22:09 GMT2023-12-05T21:22:09Z50731- Numerical methods for the generalized Zakharov systemhttps://scholarbank.nus.edu.sg/handle/10635/104826Title: Numerical methods for the generalized Zakharov system
Authors: Bao, W.; Sun, F.; Wei, G.W.
Abstract: 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 invariant 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 spectral-order 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. © 2003 Elsevier B.V. All rights reserved.
Mon, 01 Sep 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048262003-09-01T00:00:00Z
- Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensationhttps://scholarbank.nus.edu.sg/handle/10635/104828Title: Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation
Authors: Bao, W.; Jaksch, D.; Markowich, P.A.
Abstract: We study the numerical solution of the time-dependent Gross-Pitaevskii equation (GPE) describing a Bose-Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d Gross-Pitaevskii equation and obtain a four-parameter model. Identifying 'extreme parameter regimes', the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a time-splitting spectral method to discretize the time-dependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of Bose-Einstein condensation. © 2003 Elsevier Science B.V. All rights reserved.
Thu, 01 May 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048282003-05-01T00:00:00Z
- Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensionshttps://scholarbank.nus.edu.sg/handle/10635/104779Title: Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions
Authors: Bao, W.
Abstract: In this paper we design high-order (non)local artificial boundary conditions (ABCs) which are different from those proposed by Han, H. & Bao, W. (1997 Numer. Math., 77, 347-363) for incompressible materials, and present error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions. The finite-element approximation (especially its corresponding stiff matrix) becomes much simpler (sparser) when it is formulated in a bounded computational domain using the new (non)local approximate ABCs. Our error bounds indicate how the errors of the finite-element approximations depend on the mesh size, terms used in the approximate ABCs and the location of the artificial boundary. Numerical examples of the exterior Stokes equations outside a circle in the plane are presented. Numerical results demonstrate the performance of our error bounds.
Wed, 01 Jan 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047792003-01-01T00:00:00Z
- Error bounds for the finite element approximation of an incompressible material in an unbounded domainhttps://scholarbank.nus.edu.sg/handle/10635/104778Title: Error bounds for the finite element approximation of an incompressible material in an unbounded domain
Authors: Bao, W.; Han, H.
Abstract: In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds.
Wed, 01 Jan 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047782003-01-01T00:00:00Z
- Error estimates on the random projection methods for hyperbolic conservation laws with stiff reaction termshttps://scholarbank.nus.edu.sg/handle/10635/104780Title: Error estimates on the random projection methods for hyperbolic conservation laws with stiff reaction terms
Authors: Bao, W.; Jin, S.
Abstract: In this paper we give error estimates on the random projection methods, recently introduced by the authors, for numerical simulations of the hyperbolic conservation laws with stiff reaction terms: ut+f(u)x=-1(u-α)u2-1,-1
Sun, 01 Dec 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047802002-12-01T00:00:00Z
- Efficient and stable numerical methods for the generalized and vector Zakharov systemhttps://scholarbank.nus.edu.sg/handle/10635/104774Title: Efficient and stable numerical methods for the generalized and vector Zakharov system
Authors: Bao, W.; Sun, F.
Abstract: We present efficient and stable numerical methods for approximations of the generalized Zakharov system (GZS) and vector Zakharov system for multicomponent plasma (VZSM) with/without a linear damping term. The key points in the methods are based on (i) a time-splitting discretization of a Schrödinger-type equation in GZS or VZSM, (ii) discretizing a nonlinear wave-type equation by a pseudospectral method for spatial derivatives, and (iii) solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The methods are explicit, unconditionally stable, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant when there is no damping term in GZS or VZSM, conserve (or keep the same decay rate of) the wave energy as that in GZS or VZSM without a (or with a linear) damping term, and give exact results for the plane-wave solution. Extensive numerical tests are presented for plane waves and solitary-wave collisions in one-dimensional GZS, and we also give the dynamics of three-dimensional VZSM to demonstrate our new efficient and accurate numerical methods. Furthermore, the methods are applied to study the convergence and quadratic convergence rates of VZSM to GZS and of GZS to the nonlinear Schrödinger (NLS) equation in the "subsonic limit" regime (0 < ε < 1), where the parameter ε is inversely proportional to the acoustic speed. Our tests also suggest that the following meshing strategy (or ε-resolution) is admissible in this regime: spatial mesh size h = O(ε) and time step k = O(ε). © 2005 Society for Industrial and Applied Mathematics.
Sat, 01 Jan 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047742005-01-01T00:00:00Z
- Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functionalhttps://scholarbank.nus.edu.sg/handle/10635/104795Title: Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional
Authors: Bao, W.; Tang, W.
Abstract: In this paper, we propose a new numerical method to compute the ground-state solution of trapped interacting Bose-Einstein condensation 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. © 2003 Elsevier Science B.V. All rights reserved.
Thu, 01 May 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047952003-05-01T00:00:00Z
- Uniform error estimates of finite difference methods for the nonlinear schrödinger equation with wave operatorhttps://scholarbank.nus.edu.sg/handle/10635/104425Title: Uniform error estimates of finite difference methods for the nonlinear schrödinger equation with wave operator
Authors: Bao, W.; Cai, Y.
Abstract: We establish uniform error estimates of finite difference methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε (ε ε (0, 1]). When ε → 0 +, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., 0 < ε
Sun, 01 Jan 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044252012-01-01T00:00:00Z
- Dimension reduction for anisotropic Bose-Einstein condensates in the strong interaction regimehttps://scholarbank.nus.edu.sg/handle/10635/127923Title: Dimension reduction for anisotropic Bose-Einstein condensates in the strong interaction regime
Authors: Bao W.; Treust L.L.; Mehats F.
Thu, 01 Jan 2015 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1279232015-01-01T00:00:00Z
- Three-dimensional simulation of jet formation in collapsing condensateshttps://scholarbank.nus.edu.sg/handle/10635/104886Title: Three-dimensional simulation of jet formation in collapsing condensates
Authors: Bao, W.; Jaksch, D.; Markowich, P.A.
Abstract: We numerically study the behaviour of collapsing and exploding condensates using the parameters of the experiments by Donley et al (2001 Nature 412 295). 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.
Wed, 28 Jan 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048862004-01-28T00:00:00Z
- On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regimehttps://scholarbank.nus.edu.sg/handle/10635/104836Title: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Authors: Bao, W.; Jin, S.; Markowich, P.A.
Abstract: In this paper we study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant e is small. In this regime, the equation propagates oscillations with a wavelength of O (e), and finite difference approximations require the spatial mesh size h = o (e) and the time step k = o (e) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform L2-approximation of the wave function. The spectral time-splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L2-approximation of the wave function for k = o (e) and h = O (e). Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g., k independent of e, and h = O (e)) are admissible for obtaining "correct" observables. Finally, we address the application to nonlinear Schrödinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies. © 2002 Elsevier Science (USA).
Sun, 20 Jan 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048362002-01-20T00:00:00Z
- The random projection method for stiff multispecies detonation capturinghttps://scholarbank.nus.edu.sg/handle/10635/104881Title: The random projection method for stiff multispecies detonation capturing
Authors: Bao, W.; Jin, S.
Abstract: In this paper we extend the random projection method, proposed for general hyperbolic systems with stiff reaction terms, for underresolved numerical simulation of stiff, inviscid, multispecies detonation waves. The key idea in this method is to randomize the ignition temperatures in suitable domains. Several numerical experiments, in both one and two dimensions, demonstrate the reliability and robustness of this novel method. © 2002 Elsevier Science (USA).
Wed, 01 May 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048812002-05-01T00:00:00Z
- The random projection method for a model problem of combustion with stiff chemical reactionshttps://scholarbank.nus.edu.sg/handle/10635/104879Title: The random projection method for a model problem of combustion with stiff chemical reactions
Authors: Bao, W.
Abstract: In this paper we extend the random projection method, recently proposed by the author and S. Jin [J. Comput. Phys. 163 (2000) 216] for under resolved numerical simulations of a qualitative model problem for combustion with stiff chemical reactions: ut + (f(u) - q0z)x = 0, zx = 1/ε φ(u)z. In this problem, the reaction time ε is 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 is introduced recently to handle this kind of numerical difficulty. The key idea in this method is to randomize the ignition temperature in a suitable domain. Several numerical experiments demonstrate the reliability and robustness of this method. © 2002 Elsevier Science Inc. All rights reserved.
Thu, 15 Aug 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048792002-08-15T00:00:00Z
- The random projection method for stiff detonation capturinghttps://scholarbank.nus.edu.sg/handle/10635/104880Title: The random projection method for stiff detonation capturing
Authors: Bao, W.; Jin, S.
Abstract: In this paper we present a simple and robust random projection method for under-resolved numerical simulation of stiff detonation waves in chemically reacting flows. This method is based on the random projection method proposed by the authors for general hyperbolic systems with stiff reaction terms [W. Bao and S. Jin, J. Comput. Phys., 163 (2000), pp. 216-248], where the ignition temperature is randomized in a suitable domain. It is simplified using the equations of instantaneous reaction and then extended to handle the interactions of detonations. Extensive numerical experiments, including interaction of detonation waves, and in two dimensions, demonstrate the reliability and robustness of this novel method.
Tue, 01 Jan 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048802002-01-01T00:00:00Z
- Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimeshttps://scholarbank.nus.edu.sg/handle/10635/104831Title: Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes
Authors: Bao, W.; Jin, S.; Markowich, P.A.
Abstract: In this paper we study the performance of time-splitting spectral approximations for general nonlinear Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant ε is small. The time-splitting spectral approximation under study is explicit, unconditionally stable and conserves the position density in L 1. Moreover it is time-transverse invariant and time-reversible when the corresponding NLS is. Extensive numerical tests are presented for weak/strong focusing/defocusing nonlinearities, for the Gross-Pitaevskii equation, and for current-relaxed quantum hydrodynamics. The tests are geared towards the understanding of admissible meshing strategies for obtaining "correct" physical observables in the semiclassical regimes. Furthermore, comparisons between the solutions of the NLS and its hydrodynamic semiclassical limit are presented.
Mon, 01 Sep 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048312003-09-01T00:00:00Z
- On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experimentshttps://scholarbank.nus.edu.sg/handle/10635/104833Title: On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments
Authors: Bao, W.; Markowich, P.A.; Schmeiser, C.; Weishäupl, R.M.
Abstract: The three-dimensional (3D) Gross-Pitaevskii equation with strongly anisotropic confining potential is analyzed. The formal limit as the ratio of the frequencies ε tends to zero provides a denumerable system of two-dimensional Gross-Pitaevskii equations, strongly coupled through the cubic nonlinearities. To numerically solve the asymptotic approximation only a finite number of limiting equations is considered. Finally, the approximation error for a fixed number of equations is compared for different ε tending to zero. On the other hand, the approximation error for an increasing number of terms in the approximation is observed. © World Scientific Publishing Company.
Sun, 01 May 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048332005-05-01T00:00:00Z
- On the inf-sup condition of mixed finite element formulations for acoustic fluidshttps://scholarbank.nus.edu.sg/handle/10635/104834Title: On the inf-sup condition of mixed finite element formulations for acoustic fluids
Authors: Bao, W.; Wang, X.; Bathe, K.-J.
Abstract: The objective of this paper is to present a study of the solvability, stability and optimal error bounds of certain mixed finite element formulations for acoustic fluids. An analytical proof of the stability and optimal error bounds of a set of three-field mixed finite element discretizations is given, and the interrelationship between the inf-sup condition, including the numerical inf-sup test, and the eigenvalue problem pertaining to the natural frequencies is discussed.
Sun, 01 Jul 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1048342001-07-01T00:00:00Z
- Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flowhttps://scholarbank.nus.edu.sg/handle/10635/104748Title: Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow
Authors: Bao, W.; Du, Q.
Abstract: 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.
Thu, 01 Jan 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047482004-01-01T00:00:00Z
- Approximation and Comparison for Motion by Mean Curvature with Intersection Pointshttps://scholarbank.nus.edu.sg/handle/10635/104736Title: Approximation and Comparison for Motion by Mean Curvature with Intersection Points
Authors: Bao, W.
Abstract: Consider the motion of a curve in the plane under its mean curvature. It is a very interesting problem to investigate what happens when there are intersection points on the curve at which the mean curvature is singular. In this paper, we study this issue numerically by solving the Allen-Cahn equation and the nonlocal evolution equation with Kac potential. The Allen-Cahn equation is discretized by a monotone scheme, and the nonlocal evolution equation with Kac potential is discretized by the spectral method. Several curves with intersection points under motion by mean curvature are studied. From a simple analysis and our numerical results, we find that which direction to split of the curve at the intersection point depends on the angle of the curve at the point, i.e., it splits in horizontal direction when the angle α > π/2, in vertical direction when α < π/2, and in either direction when α. = π/2. © 2003 Elsevier Ltd. All rights reserved.
Wed, 01 Oct 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047362003-10-01T00:00:00Z
- An economical finite element approximation of generalized Newtonian flowshttps://scholarbank.nus.edu.sg/handle/10635/104729Title: An economical finite element approximation of generalized Newtonian flows
Authors: Bao, W.
Abstract: We consider an economical bilinear rectangular mixed finite element scheme on regular mesh for generalized Newtonian flows, where the viscosity obeys a Carreau type law for a pseudo-plastic. The key issue in the scheme is that the two components of the velocity and the pressure are defined on different meshes. Optimal error bounds for both the velocity and pressure are obtained by proving a discrete Babuška-Brezzi inf-sup condition on the regular quadrangulation. Finally, we perform some numerical experiments, including an example in a unit square with exact solutions, a backward-facing step and a four-to-one abrupt contraction generalized Newtonian flows. Numerical experiments confirm our error bounds. © 2002 Published by Elsevier Science B.V.
Fri, 21 Jun 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047292002-06-21T00:00:00Z
- An efficient and stable numerical method for the Maxwell-Dirac systemhttps://scholarbank.nus.edu.sg/handle/10635/104730Title: An efficient and stable numerical method for the Maxwell-Dirac system
Authors: Bao, W.; Li, X.-G.
Abstract: In this paper, we present an explicit, unconditionally stable and accurate numerical method for the Maxwell-Dirac system (MD) and use it to study dynamics of MD. As preparatory steps, we take the three-dimensional (3D) Maxwell-Dirac system, scale it to obtain a two-parameter model and review plane wave solution of free MD. Then we present a time-splitting spectral method (TSSP) for MD. The key point in the numerical method is based on a time-splitting discretization of the Dirac system, and to discretize nonlinear wave-type equations by pseudospectral method for spatial derivatives, and then solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals. The method is explicit, unconditionally stable, time reversible, time transverse invariant, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, it conserves the particle density exactly in discretized level and gives exact results for plane wave solution of free MD. Extensive numerical tests are presented to confirm the above properties of the numerical method. Furthermore, the tests also suggest the following meshing strategy (or E-resolution) is admissible in the 'nonrelativistic' limit regime (0 < E ≪ 1): spatial mesh size h=O(E) and time step Δt=O(E2), where the parameter E is inversely proportional to the speed of light. © 2004 Elsevier Inc. All rights reserved.
Mon, 20 Sep 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047302004-09-20T00:00:00Z
- An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearityhttps://scholarbank.nus.edu.sg/handle/10635/104731Title: An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity
Authors: Bao, W.; Jaksch, D.
Abstract: This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLSs). 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 NLSs in two dimensions with a linear, cubic, or 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 δ is larger than a threshold value δ th. We note that our method can also be applied to solve the three-dimensional Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).
Fri, 01 Aug 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047312003-08-01T00:00:00Z
- A fourth-order time-splitting laguerre-hermite pseudospectral method for bose-einstein condensateshttps://scholarbank.nus.edu.sg/handle/10635/104712Title: A fourth-order time-splitting laguerre-hermite pseudospectral method for bose-einstein condensates
Authors: Bao, W.; Shen, J.
Abstract: A fourth-order time-splitting Laguerre-Hermite pseudospectral method is introduced for Bose-Einstein condensates (BECs) in three dimensions with cylindrical symmetry. The method is explicit, time reversible, and time transverse invariant. It conserves the position density and is spectral accurate in space and fourth-order accurate in time. Moreover, the new method has two other important advantages: (i) it reduces a three-dimensional (3-D) problem with cylindrical symmetry to an effective two-dimensional (2-D) problem; (ii) it solves the problem in the whole space instead of in a truncated artificial computational domain. The method is applied to vector Gross-Pitaevskii equations (VGPEs) for multicomponent BECs. Extensive numerical tests are presented for the one-dimensional (1-D) GPE, the 2-D GPE with radial symmetry, the 3-D GPE with cylindrical symmetry, as well as 3-D VGPEs for two-component BECs, to show the efficiency and accuracy of the new numerical method. © 2005 Society for Industrial and Applied Mathematics.
Sat, 01 Jan 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047122005-01-01T00:00:00Z
- Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapsehttps://scholarbank.nus.edu.sg/handle/10635/104691Title: Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse
Authors: Bao, W.; Dong, X.; Xin, J.
Abstract: The sine-Gordon (SG) equation and perturbed nonlinear Schrödinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called light bullets) in nonlinear dispersive optical media. We begin with the (2 + 1) SG equation obtained as an asymptotic reduction in the two level dissipationless Maxwell-Bloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing-defocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet solutions of SG and perturbed NLS as well as critical NLS are reported, which validate that the solution of the perturbed NLS as well as its finite-term truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of cubic focusing NLS. In contrast, standard critical NLS is in qualitative agreement with SG only before its collapse. As a benefit of such observations, pulse propagations are studied via solving the perturbed NLS truncated by reasonably many nonlinear terms, which is a much cheaper task than solving SG equation directly. © 2010 Elsevier B.V. All rights reserved.
Thu, 01 Jul 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1046912010-07-01T00:00:00Z
- Dynamics of the ground state and central vortex states in Bose-Einstein condensationhttps://scholarbank.nus.edu.sg/handle/10635/104768Title: Dynamics of the ground state and central vortex states in Bose-Einstein condensation
Authors: Bao, W.; Zhang, Y.
Abstract: In this paper, we study dynamics of the ground state and central vortex states in Bose-Einstein condensation (BEC) analytically and numerically. We show how to define the energy of the Thomas-Fermi (TF) approximation, prove that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in linear case, derive a second-order ordinary differential equation (ODE) which shows that time-evolution of the condensate width is a periodic function with/without a perturbation by using the variance identity, prove that the angular momentum expectation is conserved in two dimensions (2D) with a radial symmetric trap and 3D with a cylindrical symmetric trap for any initial data, and study numerically stability of central vortex states as well as interaction between a few central vortices with winding numbers ±1 by a fourth-order time-splitting sine-pseudospectral (TSSP) method. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density, performs spectral accuracy for spatial derivatives and fourth-order accuracy for time derivative, and possesses "optimal" spatial/temporal resolution in the semiclassical regime. Finally we find numerically the critical angular frequency for single vortex cycling from the ground state under a far-blue detuned Gaussian laser stirrer in strong repulsive interaction regime and compare our numerical results with those in the literatures. © World Scientific Publishing Company.
Thu, 01 Dec 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1047682005-12-01T00:00:00Z
- Ground states of two-component Bose-Einstein condensates with an internal atomic josephson junctionhttps://scholarbank.nus.edu.sg/handle/10635/103356Title: Ground states of two-component Bose-Einstein condensates with an internal atomic josephson junction
Authors: Bao, W.; Cai, Y.
Abstract: In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.© 2011 Global-Science Press.
Tue, 01 Feb 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033562011-02-01T00:00:00Z
- Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensionshttps://scholarbank.nus.edu.sg/handle/10635/103540Title: Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions
Authors: Cai, Y.; Rosenkranz, M.; Lei, Z.; Bao, W.
Abstract: 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. © 2010 The American Physical Society.
Tue, 26 Oct 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035402010-10-26T00:00:00Z
- A variational-difference numerical method for designing progressive-addition lenseshttps://scholarbank.nus.edu.sg/handle/10635/102785Title: A variational-difference numerical method for designing progressive-addition lenses
Authors: Jiang, W.; Bao, W.; Tang, Q.; Wang, H.
Abstract: We propose a variational-difference method for designing the optical free form surface of progressive-addition lenses (PALs). The PAL, which has a front surface with three important zones including the far-view, near-view and intermediate zones, is often used to remedy presbyopia by distributing optical powers of the three zones progressively and smoothly. The problem for designing PALs could be viewed as a functional minimization problem. Compared with the existing literature which solved the problem by the B-spline finite element method, the essence of the proposed variational-difference numerical method lies in minimizing the functional directly by finite difference method and/or numerical quadratures rather than in approximating the solution of the corresponding Euler-Lagrange equation to the functional. It is very easily understood and implemented by optical engineers, and the numerical results indicate that it can produce satisfactory designs for optical engineers in several seconds. We believe that our method can be a powerful candidate tool for designing various specifications of PALs. © 2013 Published by Elsevier Ltd. All rights reserved.
Wed, 01 Jan 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027852014-01-01T00:00:00Z
- A time-splitting spectral method for three-wave interactions in media with competing quadratic and cubic nonlinearitieshttps://scholarbank.nus.edu.sg/handle/10635/102777Title: A time-splitting spectral method for three-wave interactions in media with competing quadratic and cubic nonlinearities
Authors: Bao, W.; Zheng, C.
Abstract: This paper introduces an extension of the time-splitting spectral (TSSP) method for solving a general model of three-wave optical interactions, which typically arises from norlinear optics, when the transmission media has competing quadratic and cubic nonlinearities. The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way, which allows us to develop an explicit and unconditionally stable numerical method for the problem. Furthermore, the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction, is time reversible and time transverse invariant, and conserves the total wave energy (or power or the norm of the solutions) in discretized level. Numerical examples are presented to demonstrate the efficiency and high resolution of the method. Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension (1D) and 2D. © 2007 Global-Science Press.
Thu, 01 Feb 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027772007-02-01T00:00:00Z
- A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problemshttps://scholarbank.nus.edu.sg/handle/10635/102779Title: A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problems
Authors: Bao, W.; Chai, M.-H.
Abstract: In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are presented to locate the positions and characterize the widths of boundary layers and/or interior layers in the solution. A uniformly convergent numerical method is proposed by using the normalized gradient flow and piecewise uniform mesh techniques based on the asymptotic approximations for the problem. Extensive numerical results are reported to demonstrate the effectiveness of our numerical method for the problems. Finally, the method is applied to compute ground and excited states of Bose-Einstein condensation in the semiclassical regime and some conclusive findings are reported. © 2008 Global-Science Press.
Tue, 01 Jul 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027792008-07-01T00:00:00Z
- A generalized-laguerre-fourier-hermite pseudospectral method for computing the dynamics of rotating bose-einstein condensateshttps://scholarbank.nus.edu.sg/handle/10635/102660Title: A generalized-laguerre-fourier-hermite pseudospectral method for computing the dynamics of rotating bose-einstein condensates
Authors: Bao, W.; Li, H.; Shen, J.
Abstract: A time-splitting generalized-Laguerre-Fourier-Hermite pseudospectral method is proposed for computing the dynamics of rotating Bose-Einstein condensates (BECs) in two and three dimensions. The new numerical method is based on the following: (i) the use of a timesplitting technique for decoupling the nonlinearity; (ii) the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient; and (iii) the construction of eigenfunctions for the linear operator by properly scaling the generalized-Laguerre, Fourier, and Hermite functions. The new method enjoys the following properties: (i) it is explicit, time reversible, and time transverse invariant; (ii) it conserves the position density and is spectrally accurate in space and second-order or fourth-order accurate in time; and (iii) it solves the problem in the original whole space instead of in a truncated artificial computational domain. The method is also extended to solve the coupled Gross-Pitaevskii equations for the dynamics of rotating two-component and spin-1 BECs. Extensive numerical results for the dynamics of BECs are reported to demonstrate the accuracy and efficiency of the new method for rotating BECs. © 2009 Society for Industrial and Applied Mathematics.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026602009-01-01T00:00:00Z
- A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensateshttps://scholarbank.nus.edu.sg/handle/10635/102661Title: A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates
Authors: Bao, W.; Shen, J.
Abstract: 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. © 2008 Elsevier Inc. All rights reserved.
Mon, 01 Dec 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026612008-12-01T00:00:00Z
- An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensateshttps://scholarbank.nus.edu.sg/handle/10635/102827Title: An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates
Authors: Bao, W.; Wang, H.
Abstract: In this paper, we propose an efficient and spectrally accurate numerical method for computing the dynamics of rotating Bose-Einstein condensates (BEC) in two dimensions (2D) and 3D based on the Gross-Pitaevskii equation (GPE) with an angular momentum rotation term. By applying a time-splitting technique for decoupling the nonlinearity and properly using the alternating direction implicit (ADI) technique for the coupling in the angular momentum rotation term in the GPE, at every time step, the GPE in rotational frame is decoupled into a nonlinear ordinary differential equation (ODE) and two partial differential equations with constant coefficients. This allows us to develop new time-splitting spectral methods for computing the dynamics of BEC in a rotational frame. The new numerical method is explicit, unconditionally stable, and of spectral accuracy in space and second-order accuracy in time. Moreover, it is time reversible and time transverse invariant, and conserves the position density in the discretized level if the GPE does. Extensive numerical results are presented to confirm the above properties of the new numerical method for rotating BEC in 2D and 3D. © 2006 Elsevier Inc. All rights reserved.
Wed, 20 Sep 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028272006-09-20T00:00:00Z
- An exponential wave integrator sine pseudospectral method for the klein-gordon-zakharov systemhttps://scholarbank.nus.edu.sg/handle/10635/102835Title: An exponential wave integrator sine pseudospectral method for the klein-gordon-zakharov system
Authors: Bao, W.; Dong, X.; Zhao, X.
Abstract: An exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0 > e ≤ 1 and 0 > γ = 1 which are inversely proportional to the plasma frequency and the speed of sound, respectively. The main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space. The method is explicit, symmetric in time, and it is of spectral accuracy in space and second-order accuracy in time for any fixed e = e0 and γ = γ0. In the O(1)-plasma frequency and speed of sound regime, i.e., e = O(1) and γ = O(1), we establish rigorously error estimates for the numerical method in the energy space H1 × L2. We also study numerically the resolution of the method in the simultaneous high-plasma-frequency and subsonic limit regime, i.e., (e, γ) - 0 under e ≥ γ. In fact, in this singular limit regime, the solution of the KGZ system is highly oscillating in time, i.e., there are propagating waves with wavelength of O(e2) and O(1) in time and space, respectively. Our extensive numerical results suggest that, in order to compute "correct" solutions in the simultaneous high-plasma-frequency and subsonic limit regime, the meshing strategy (or e-scalability) is time step t = O(e2) and mesh size h = O(1) independent of e. Finally, we also observe numerically that the method has the property of near conservation of the energy over long time in practical computations.Copyright © by SIAM.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028352013-01-01T00:00:00Z
- Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regimehttps://scholarbank.nus.edu.sg/handle/10635/102852Title: Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime
Authors: Bao, W.; Dong, X.
Abstract: We analyze rigourously error estimates and compare numerically temporal/spatial resolution of various numerical methods for solving the Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter 0 < e{open} ≪ 1 which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time, i. e. there are propagating waves with wavelength of O(e{open} 2) and O(1) in time and space, respectively. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter e{open}. Based on the error bounds, in order to compute 'correct' solutions when 0 < e{open} ≪ 1the four FDTD methods share the same e{open}scalability: τ = O (e{open} 3). Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their e{open}-scalability is improved to τ = O(1) and τ = O (e{open} 2) for linear and nonlinear KG equations, respectively, when 0 < e{open} ≪ 1 Numerical results are reported to support our error estimates. © 2011 Springer-Verlag.
Wed, 01 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028522012-02-01T00:00:00Z
- Numerical study of quantized vortex interaction in the ginzburg-landau equation on bounded domainshttps://scholarbank.nus.edu.sg/handle/10635/103657Title: Numerical study of quantized vortex interaction in the ginzburg-landau equation on bounded domains
Authors: Bao, W.; Tang, Q.
Abstract: In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE)with a dimensionless parameter ε>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a reviewof the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains. © 2013 Global-Science Press.
Sun, 01 Sep 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036572013-09-01T00:00:00Z
- Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson starshttps://scholarbank.nus.edu.sg/handle/10635/103649Title: Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars
Authors: Bao, W.; Dong, X.
Abstract: Efficient and accurate numerical methods are presented for computing ground states and dynamics of the three-dimensional (3D) nonlinear relativistic Hartree equation both without and with an external potential. This equation was derived recently for describing the mean field dynamics of boson stars. In its numerics, due to the appearance of pseudodifferential operator which is defined in phase space via symbol, spectral method is more suitable for the discretization in space than other numerical methods such as finite difference method, etc. For computing ground states, a backward Euler sine pseudospectral (BESP) method is proposed based on a gradient flow with discrete normalization; and respectively, for computing dynamics, a time-splitting sine pseudospectral (TSSP) method is presented based on a splitting technique to decouple the nonlinearity. Both BESP and TSSP are efficient in computation via discrete sine transform, and are of spectral accuracy in spatial discretization. TSSP is of second-order accuracy in temporal discretization and conserves the normalization in discretized level. In addition, when the external potential and initial data for dynamics are spherically symmetric, the original 3D problem collapses to a quasi-1D problem, for which both BESP and TSSP methods are extended successfully with a proper change of variables. Finally, extensive numerical results are reported to demonstrate the spectral accuracy of the methods and to show very interesting and complicated phenomena in the mean field dynamics of boson stars. © 2011 Elsevier Inc.
Fri, 10 Jun 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036492011-06-10T00:00:00Z
- Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equationhttps://scholarbank.nus.edu.sg/handle/10635/103648Title: Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation
Authors: Bao, W.; Tang, Q.; Xu, Z.
Abstract: In this paper, we propose new efficient and accurate numerical methods for computing dark solitons and review some existing numerical methods for bright and/or dark solitons in the nonlinear Schrödinger equation (NLSE), and compare them numerically in terms of accuracy and efficiency. We begin with a review of dark and bright solitons of NLSE with defocusing and focusing cubic nonlinearities, respectively. For computing dark solitons, to overcome the nonzero and/or non-rest (or highly oscillatory) phase background at far field, we design efficient and accurate numerical methods based on accurate and simple artificial boundary conditions or a proper transformation to rest the highly oscillatory phase background. Stability and conservation laws of these numerical methods are analyzed. For computing interactions between dark and bright solitons, we compare the efficiency and accuracy of the above numerical methods and different existing numerical methods for computing bright solitons of NLSE, and identify the most efficient and accurate numerical methods for computing dark and bright solitons as well as their interactions in NLSE. These numerical methods are applied to study numerically the stability and interactions of dark and bright solitons in NLSE. Finally, they are extended to solve NLSE with general nonlinearity and/or external potential and coupled NLSEs with vector solitons. © 2012 Elsevier Inc.
Fri, 15 Feb 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036482013-02-15T00:00:00Z
- Numerical methods for computing the ground state of spin-1 Bose-Einstein condensates in a uniform magnetic fieldhttps://scholarbank.nus.edu.sg/handle/10635/103650Title: Numerical methods for computing the ground state of spin-1 Bose-Einstein condensates in a uniform magnetic field
Authors: Lim, F.Y.; Bao, W.
Abstract: 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. © 2008 The American Physical Society.
Mon, 01 Dec 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036502008-12-01T00:00:00Z
- Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equationhttps://scholarbank.nus.edu.sg/handle/10635/103653Title: Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation
Authors: Zhang, Y.; Bao, W.; Du, Q.
Abstract: The rich dynamics of quantized vortices governed by the Ginzburg-Landau-Schrödinger equation (GLSE) is an interesting problem studied in many application fields. Although recent mathematical analysis and numerical simulations have led to a much better understanding of such dynamics, many important questions remain open. In this article, we consider numerical simulations of the GLSE in two dimensions with non-zero far-field conditions. Using two-dimensional polar coordinates, transversely highly oscillating far-field conditions can be efficiently resolved in the phase space, thus giving rise to an unconditionally stable, efficient and accurate time-splitting method for the problem under consideration. This method is also time reversible for the case of the non-linear Schrödinger equation. By applying this numerical method to the GLSE, we obtain some conclusive experimental findings on issues such as the stability of quantized vortex, interaction of two vortices, dynamics of the quantized vortex lattice and the motion of vortex with an inhomogeneous external potential. Discussions on these simulation results and the recent theoretical studies are made to provide further understanding of the vortex stability and vortex dynamics described, by the GLSE. © 2007 Cambridge University Press.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036532007-01-01T00:00:00Z
- Mathematical theory and numerical methods for Bose-Einstein condensationhttps://scholarbank.nus.edu.sg/handle/10635/103528Title: Mathematical theory and numerical methods for Bose-Einstein condensation
Authors: Bao, W.; Cai, Y.
Abstract: 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. © American Institute of Mathematical Sciences.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035282013-01-01T00:00:00Z
- Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical lawhttps://scholarbank.nus.edu.sg/handle/10635/212386Title: Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law
Authors: Xu, Z.; Bao, W.; Shi, S.
Abstract: We study analytically and numerically stability and interaction patterns of quantized vortex lattices governed by the reduced dynamical law – a system of ordinary differential equations (ODEs) – in superconductivity. By deriving several non-autonomous first integrals of the ODEs, we obtain qualitatively dynamical properties of a cluster of quantized vortices, including global existence, finite time collision, equilibrium solution and invariant solution manifolds. For a vortex lattice with 3 vortices, we establish orbital stability when they have the same winding number and find different collision patterns when they have different winding numbers. In addition, under several special initial setups, we can obtain analytical solutions for the nonlinear ODEs. © 2018 American Institute of Mathematical Sciences. All Rights Reserved.
Mon, 01 Jan 2018 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2123862018-01-01T00:00:00Z
- Subdiffusive spreading of a Bose-Einstein condensate in random potentialshttps://scholarbank.nus.edu.sg/handle/10635/104213Title: Subdiffusive spreading of a Bose-Einstein condensate in random potentials
Authors: Min, B.; Li, T.; Rosenkranz, M.; Bao, W.
Abstract: 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 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. © 2012 American Physical Society.
Wed, 14 Nov 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042132012-11-14T00:00:00Z
- The dynamics and interaction of quantized vortices in the Ginzburg-landau-schrödinger equationhttps://scholarbank.nus.edu.sg/handle/10635/104284Title: The dynamics and interaction of quantized vortices in the Ginzburg-landau-schrödinger equation
Authors: Zhang, Y.; Bao, W.; Du, Q.
Abstract: The dynamic laws of quantized vortex interactions in the Ginzburg-Landau-Schrödinger equation (GLSE) are analytically and numerically studied. A review of the reduced dynamic laws governing the motion of vortex centers in the GLSE is provided. The reduced dynamic laws are solved analytically for some special initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently in [Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau- Schrödinger equation, European J. Appl. Math., to appear], we can qualitatively and quantitatively compare quantized vortex interaction patterns of the GLSE with those from the reduced dynamic laws. Some conclusive findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex interactions in the GLSE. Finally, the vortex motion under an inhomogeneous potential in the GLSE is also studied. © 2007 Society for Industrial and Applied Mathematics.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042842007-01-01T00:00:00Z
- Scattering and bound states in two-dimensional anisotropic potentialshttps://scholarbank.nus.edu.sg/handle/10635/104077Title: Scattering and bound states in two-dimensional anisotropic potentials
Authors: Rosenkranz, M.; Bao, W.
Abstract: 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. © 2011 American Physical Society.
Tue, 22 Nov 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040772011-11-22T00:00:00Z
- Singular limits of Klein Gordon Schrödinger equations to Schrödinger Yukawa equationshttps://scholarbank.nus.edu.sg/handle/10635/104125Title: Singular limits of Klein Gordon Schrödinger equations to Schrödinger Yukawa equations
Authors: Bao, W.; Dong, X.; Wang, S.
Abstract: In this paper, we study analytically and numerically the singular limits of the nonlinear Klein-Gordon-Schrödinger (KGS) equations in ℝd (d = 1, 2, 3) both with and without a damping term to the nonlinear Schrödinger-Yukawa (SY) equations. By using the two-scale matched asymptotic expansion, formal limits of the solution of the KGS equations to the solution of the SY equations are derived with an additional correction in the initial layer. Then for general initial data, weak and strong convergence results are established for the formal limits to provide rigorous mathematical justification for the matched asymptotic approximation by using the weak compactness argument and the (modulated) energy method, respectively. In addition, for well-prepared initial data, optimal quadratic and linear convergence rates are obtained for the KGS equations both with and without the damping term, respectively, and for ill-prepared initial data, the optimal linear convergence rate is obtained. Finally, numerical results for the KGS equations are presented to confirm the asymptotic and analytic results. © 2010 Society for Industrial and Applied Mathematics.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041252010-01-01T00:00:00Z
- Self-trapping of Bose-Einstein condensates expanding into shallow optical latticeshttps://scholarbank.nus.edu.sg/handle/10635/104090Title: Self-trapping of Bose-Einstein condensates expanding into shallow optical lattices
Authors: Rosenkranz, M.; Jaksch, D.; Yin Lim, F.; Bao, W.
Abstract: We observe a sudden breakdown of the transport of a strongly repulsive Bose-Einstein condensate through a shallow optical lattice of finite width. We are able to attribute this behavior to the development of a self-trapped state by using accurate numerical methods and an analytical description in terms of nonlinear Bloch waves. The dependence of the breakdown on the lattice depth and the interaction strength is investigated. We show that it is possible to prohibit the self-trapping by applying a constant offset potential to the lattice region. Furthermore, we observe the disappearance of the self-trapped state after a finite time as a result of the revived expansion of the condensate through the lattice. This revived expansion is due to the finite width of the lattice. © 2008 The American Physical Society.
Wed, 11 Jun 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040902008-06-11T00:00:00Z
- Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive couplinghttps://scholarbank.nus.edu.sg/handle/10635/104091Title: Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling
Authors: Bruderer, M.; Bao, W.; Jaksch, D.
Abstract: 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 β*. 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. © 2008 Europhysics Letters Association.
Thu, 01 May 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040912008-05-01T00:00:00Z
- Quantized vortex stability and interaction in the nonlinear wave equationhttps://scholarbank.nus.edu.sg/handle/10635/104004Title: Quantized vortex stability and interaction in the nonlinear wave equation
Authors: Bao, W.; Zeng, R.; Zhang, Y.
Abstract: The stability and interaction of quantized vortices in the nonlinear wave equation (NLWE) are investigated both numerically and analytically. A review of the reduced dynamic law governing the motion of vortex centers in the NLWE is provided. The second order nonlinear ordinary differential equations for the reduced dynamic law are solved analytically for some special initial data. Using 2D polar coordinates, the transversely highly oscillating far field conditions can be efficiently resolved in the phase space, thus giving rise to an efficient and accurate numerical method for the NLWE with non-zero far field conditions. By applying this numerical method to the NLWE, we study the stability of quantized vortices and find numerically that the quantized vortices with winding number m = ± 1 are dynamically stable, and resp. | m | > 1 are dynamically unstable, in the dynamics of NLWE. We then compare numerically quantized vortex interaction patterns of the NLWE with those from the reduced dynamic law qualitatively and quantitatively. Some conclusive findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex stability and interactions in the NLWE. Finally, the vortex motion under an inhomogeneous potential in the NLWE is also studied. © 2008 Elsevier B.V. All rights reserved.
Wed, 01 Oct 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040042008-10-01T00:00:00Z
- Phase field approach for simulating solid-state dewetting problemshttps://scholarbank.nus.edu.sg/handle/10635/103934Title: Phase field approach for simulating solid-state dewetting problems
Authors: Jiang, W.; Bao, W.; Thompson, C.V.; Srolovitz, D.J.
Abstract: We propose a phase field model for simulating solid-state dewetting and the morphological evolution of patterned islands on a substrate. The evolution is governed by the Cahn-Hilliard equation with isotropic surface tension and variable scalar mobility. The proposed approach easily deals with the complex boundary conditions arising in the solid-state dewetting problem. Since the method does not explicitly track the moving surface, it naturally captures the topological changes that occur during film/island morphology evolution. The numerical method is based on the cosine pseudospectral method together with a highly efficient, stabilized, semi-implicit algorithm. Numerical results on solid-state dewetting in two dimensions demonstrate the excellent performance of the method, including stability, accuracy and numerical efficiency. The method was easily extended to three dimensions (3D), with no essential difference from the two-dimensional algorithm. Numerical experiments in 3D demonstrate the ability of the model to capture many of the complexities that have been observed in the experimental dewetting of thin films on substrates and the evolution of patterned islands on substrates. © 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Sat, 01 Sep 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039342012-09-01T00:00:00Z