Efficient numerical methods for computing ground states and dynamics of dipolar Bose–Einstein condensates

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.     

On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system

In this paper, we deal with the computation of ground state and dynamics of the Schrödinger–Poisson–Slater (SPS) system. To this end, backward Euler and time-splitting pseudospectral methods are proposed for the nonlinear Schrödinger equation with the nonlocal Hartree potential approximated by solving a Poisson equation. The approximation approaches for the Hartree potential include fast convolution algorithms, which are accelerated by using FFT in 1D and fast multipole method (FMM) in 2D and 3D, and sine/Fourier pseudospectral methods. The inconsistency in 0-mode in Fourier pseudospectral approach is pointed out, which results in a significant loss of high-order of accuracy as expected for spectral methods. Numerical comparisons show that in 1D the fast convolution and sine pseudospectral approaches are compatible. While, in 3D the fast convolution approach based on FMM is second-order accurate and the Fourier pseudospectral approach is better than it from both efficiency and accuracy point of view. Among all these approaches, the sine pseudospectral one is the best candidate in the numerics of the SPS system. Finally, we apply the backward Euler and time-splitting sine pseudospectral methods to study the ground state and dynamics of 3D SPS system in different setups.     

Explicit multi-symplectic method for the Zakharov—Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.     

A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrödinger–Poisson–Slater system

In a recent paper we proposed and compared various approaches to compute the ground state and dynamics of the Schrödinger–Poisson–Slater (SPS) system for general external potential and initial condition, concluding that the methods based on sine pseudospectral discretization in space are the best candidates. This note is concerned with the case that the external potential and initial condition are spherically symmetric. For the SPS system with spherical symmetry, via applying a proper change of variables into the reduced quasi-1D model we simplify the methods proposed for the general 3D case such that both the memory and computational load are significantly reduced.     

Modified Structure-Preserving Schemes for the Degasperis-Procesi Equation

We investigate the structure-preserving numerical algorithm of the Degasperis-Procesi equation which can be transformed into a bi-Hamiltonian form using the discrete variational derivative method.Based on two different space discretization methods,the Fourier pseudospectral method and the wavelet collocation method,we develop two modified structure-preserving schemes under the periodic boundary condition.These proposed schemes are proved to preserve the Hamiltonian invariants theoretically and numerically.Meanwhile,the numerical results confirm that they can simulate the propagation of solitons effectively for a long time.     

Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method

The Schur-decomposition for three-dimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3D Schur-decomposition solver is the least, and almost only one thirtieth to one fiftieth CPU time is needed when using the spectral method with 3D Schur-decomposition solver compared with the standard discrete ordinates method.     

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach     

Fourier拟谱配点方法对第一类边界积分方程的应用

本文用Fourier拟谱配点方法求解有广泛应用的以对数核为主部的第一类边界积分方程,文中通过对积分算子的象征作拟谱插值来建立近似方程,利用快速Fourier变换将计算切换到频率空间进行。本文计算结果表明,用上述拟谱配点方法计算的数值精度较Galerkin配点法更为满意。     

Generalized Pseudospectral Method for Solving the Time-Dependent Schrödinger Equation Involving the Coulomb Potential

We present an accurate and efficient generalized pseudospectral method for solving the time-dependent Schrödinger equation for atomic systems interacting with intense laser fields. In this method, the time propagation of the wave function is calculated using the well-known second-order split-operator method implemented by the numerically exact, fast transform between the grid and spectral representations. In the grid representation, the radial coordinate is discretized using the Coulomb wave discrete variable representation (CWDVR), and the angular dependence of the wave function is expanded in the Gauss-Legendre-Fourier grid. In the spectral representation, the wave function is expanded in terms of the eigenfunctions of the field-free zero-order Hamiltonian. Calculations on the high order harmonic generation and ionization dynamics of hydrogen atom in strong laser pulses are presented to demonstrate the accuracy and efficiency of the present method. This new algorithm will be found more computationally attractive than the close-coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.     

A new complex variable meshless method for transient heat conduction problems

<正>In this paper,based on the improved complex variable moving least-square(ICVMLS) approximation,a new complex variable meshless method(CVMM) for two-dimensional(2D) transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations,and the essential boundary conditions are imposed by the penalty method.As the transient heat conduction problems are related to time,the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization.Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained.In order to demonstrate the applicability of the proposed method,numerical examples are given to show the high convergence rate,good accuracy,and high efficiency of the CVMM presented in this paper.     

基于高斯伪谱方法的混沌系统最优控制

针对混沌系统最优控制问题,提出一种基于高斯伪谱方法的数值求解新算法. 首先在勒让德-高斯点上构造Lagrange插值多项式并近似表示混沌系统最优控制中的状态变量和控制变量;接着将连续空间的最优控制问题转化为非线性规划问题;然后通过序列二次规划（SQP）算法获得最优解;最后对三个典型混沌系统的仿真实验结果表明,新方法能有效和快速地实现混沌系统的最优控制. 关键词： 混沌系统 最优控制 高斯伪谱法 非线性规划     

强磁场中氢原子的能级结构

发展了一套简单高效的非微扰理论方法研究强磁场中的原子能级结构.作为例子,给出了磁场强度从0到1000个原子单位,氢原子基态和低激发态的结合能以及四极矩等重要原子能级结构参数.结果表明:相对于其他的高精度计算方法,该方法不仅能计算出高精度的能级位置而且可方便给出精确的电子波函数.此外,该法还具有很强的普适性,可直接推广到原子与任意方向的交叉电磁场相互作用的研究. 关键词： 强磁场 CWDVR谱方法 氢原子能级结构 四极矩     

An efficient numerical method for computing dynamics of spin F = 2 Bose–Einstein condensates

In this paper, we extend the efficient time-splitting Fourier pseudospectral method to solve the generalized Gross–Pitaevskii (GP) equations, which model the dynamics of spin F = 2 Bose–Einstein condensates at extremely low temperature. Using the time-splitting technique, we split the generalized GP equations into one linear part and two nonlinear parts: the linear part is solved with the Fourier pseudospectral method; one of nonlinear parts is solved analytically while the other one is reformulated into a matrix formulation and solved by diagonalization. We show that the method keeps well the conservation laws related to generalized GP equations in 1D and 2D. We also show that the method is of second-order in time and spectrally accurate in space through a one-dimensional numerical test. We apply the method to investigate the dynamics of spin F = 2 Bose–Einstein condensates confined in a uniform/nonuniform magnetic field.     

The generalized pseudospectral approach to the bound states of the Hulthén and the Yukawa potentials

The generalized pseudospectral (GPS) method is employed to calculate the bound states of the Hulthén and the Yukawa potentials in quantum mechanics, with special emphasis onhigher excited states andstronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a non-uniform and optimal spatial discretization of the radial Schrödinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials withn <- 10 for arbitrary values of the screening parameters covering a wide range of interaction. Furthermore, excited states as high asn = 17 have been computed with good accuracy for both these potentials. Excellent agreement with the available literature data has been observed in all cases. Then > 6 states of the Yukawa potential has been considerably improved over all other existing results currently available, while the same for Hulthén potential are reported here for the first time. Excepting the 1s and 2s states of the Yukawa potential, the present method surpasses the accuracy of all other existing results in the stronger coupling region for all other states of both these systems. This offers a simple and efficient scheme for the accurate calculation of these and other screened Coulomb potentials.     

A numerical method for incompressible viscous flow simulation

We describe a numerical scheme for computing time-dependent solutions of the incompressible Navier-Stokes equations in the primitive variable formulation. This scheme uses finite elements for the space discretization and operator splitting techniques for the time discretization. The resulting discrete equations are solved using specialized nonlinear optimization algorithms that are computationally efficient and have modest storage requirements. The basic numerical kernel is the preconditioned conjugate gradient method for symmetric, positive-definite, sparse matrix systems, which can be efficiently implemented on the architectures of vector and parallel processing supercomputers.     

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional S_N discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.     

基于局部基本解法的静电场仿真分析

将局部基本解方法应用于静电场问题的模拟与分析。局部基本解方法是利用控制方程的基本解,基于局部理论和移动最小二乘原理提出的一种无网格算法。相比于有限元和有限差分等传统网格类方法,该方法仅需离散节点,避免了复杂的网格剖分难题。作为一种半解析数值技术,物理问题的基本解被作为插值基函数建立数值离散模型,从而保证了算法的较高精度。此外,与具有全局离散格式的无网格方法相比,局部基本解法更适用于高维复杂几何和大尺度模拟。二维和三维数值试验表明,该方法具有实施方便灵活,计算精度高和计算速度快等优势。为静电场仿真研究开辟新的途径,拓展了局部基本解方法的应用领域。     

Efficient spectral and pseudospectral algorithms for 3D simulations of whistler-mode waves in a plasma

Efficient spectral and pseudospectral algorithms for simulation of linear and nonlinear 3D whistler waves in a cold electron plasma are developed. These algorithms are applied to the simulation of whistler waves generated by loop antennas and spheromak-like stationary waves of considerable amplitude. The algorithms are linearly stable and show good stability properties for computations of nonlinear waves over tens of thousands of time steps. Additional speedups by factors of 10–20 (comparing single core CPU and one GPU) are achieved by using graphics processors (GPUs), which enable efficient numerical simulation of the wave propagation on relatively high resolution meshes (tens of millions nodes) in personal computing environment. Comparisons of the numerical results with analytical solutions and experiments show good agreement. The limitations of the codes and the performance of the GPU computing are discussed.     

Pseudospectral method with symplectic algorithm for the solution of time-dependent SchrSdinger equations

A pseudospectral method with symplectic algorithm for the solution of time-dependent Schrodinger equations （TDSE） is introduced. The spatial part of the wavefunction is discretized into sparse grid by pseudospectral method and the time evolution is given in symplectic scheme. This method allows us to obtain a highly accurate and stable solution of TDSE. The effectiveness and efficiency of this method is demonstrated by the high-order harmonic spectra of one-dimensional atom in strong laser field as compared with previously published work. The influence of the additional static electric field is also investigated.     

空间差分格式对有限体积法数值散射的影响

数值散射是辐射传递方程近似算法中最常见的离散误差。本文主要讨论空间差分格式对有限体积法数值散射的影响。构造激光平行及倾斜入射的物理模型,验证和比较阶梯格式、中心差分格式及指数格式下温度场的计算精度及数值散射特性。计算结果表明,在激光平行入射与倾斜入射两种情况下,阶梯格式引起的的数值散射比菱形格式及指数格式要多,但其计算精度高于菱形及指数格式。不同激光入射条件下,各种差分格式表现出的数值散射分布有明显的差异。