In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F≥3) BEC are discussed.

In this paper, we propose an efficient and accurate method to compute the ground state of 2D/3D rotating dipolar BEC by incorporating the Kernel Truncation Method (KTM) for Dipole-Dipole Interaction (DDI) evaluation into the newly-developed Preconditioned Conjugate Gradient (PCG) method [9]. Adaptation details of KTM and PCG, including multidimensional discrete convolution acceleration for KTM, choice of the preconditioners in PCG, are provided. The performance of our method is confirmed with extensive numerical tests, with emphasis on spectral accuracy of KTM and efficiency of ground state computation with PCG. Application of our method shows some interesting vortex lattice patterns in 2D and 3D respectively.

We propose a hybrid method which combines the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number ε→0, the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an O(1) difference between the solutions of the Schrödinger equation and its Dirac approximation.

In this paper, we present a numerical scheme for the steady-state radiative transfer equation (RTE) with Henyey-Greenstein scattering kernel. The scattering kernel is anisotropic but not highly forward peaked. On the one hand, for the velocity discretization, we approximate the anisotropic scattering kernel by a discrete matrix that can preserve the diffusion limit. On the other hand, for the space discretization, a uniformly convergent scheme up to the boundary or interface layer is proposed. The idea is that we first approximate the scattering coefficients as well as source by piecewise constant functions, then, in each cell, the true solution is approximated by the summation of a particular solution and a linear combinations of general solutions to the homogeneous RTE. Second-order accuracy can be observed, uniformly with respect to the mean free path up to the boundary and interface layers. The scheme works well for heterogenous medium, anisotropic sources as well as for the strong source regime.

This paper focuses on the simulation of nonlocal wave propagations in unbounded multi-scale mediums. To this end, we consider two issues: (a) the design of artificial/absorbing boundary conditions; and (b) the construction of an asymptotically compatible (AC) scheme for the nonlocal operator with general kernels. The design of ABCs facilitates us to reformulate unbounded domain problems into bounded domain problems. The construction of AC scheme facilitates us to simulate nonlocal wave propagations in multi-scale mediums. By applying the proposed ABCs and the proposed AC scheme, we investigate different wave propagation behaviors in the "local" and nonlocal mediums through numerical examples.

We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a low-dimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schrödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.

We aim at a fast finite element method for the Poisson equation in three-dimensional infinite domains. Both the exterior and strip-tail problems are considered. By introducing a suitable artificial boundary and imposing the exact boundary condition of Dirichlet-to-Neumann (DtN) type, we reduce the original infinite domain problem into a truncated finite domain problem. The point is how to efficiently implement this exact artificial boundary condition. The traditional modal expansion method is hard to apply for the strip-tail problem with a general cross section. We develop a fast algorithm based on the Padé approximation for the square root function involved in the exact artificial boundary condition. The most remarkable advantage of our method is that it is unnecessary to compute the full eigen system associated with the Laplace-Beltrami operator on the artificial boundary. Besides, compared with the modal expansion method, the computational cost of the DtN mapping is significantly reduced. We perform a complete numerical analysis on the fast algorithm. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.

We study analytically the existence and uniqueness of the ground state of the nonlinear Schrödinger equation (NLSE) with a general power nonlinearity described by the power index σ≥0. For the NLSE under a box or a harmonic potential, we can derive explicitly the approximations of the ground states and their corresponding energy and chemical potential in weak or strong interaction regimes with a fixed nonlinearity σ. Besides, we study the case where the nonlinearity σ→∞ with a fixed interaction strength. In particular, a bifurcation in the ground states is observed. Numerical results in 1D and 2D will be reported to support our asymptotic results.

This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite points. In this work, two Hermite-type functions are employed to serve as basis functions. Our main task is to find corresponding differentiation matrices which are computed recursively. Two important issues relevant to condition numbers and scaling factors will be discussed. Applications of the spectral collocation methods to multi-term fractional PDEs are also presented. Several numerical examples are carried out to demonstrate the effectiveness of the proposed methods.

In this paper, we propose using the tailored-finite-point method (TFPM) to solve the resulting parabolic or elliptic equations when minimizing the Rician denoising model developed by Getreuer et al. in [10] using augmented Lagrangian methods (ALM). Different from traditional finite difference schemes, TFPM employs the method of weighted residuals with collocation technique, which helps get more accurate approximate solutions to the equations and thus reserve more details in restored images. Numerical experiments demonstrate that with the new schemes the quality of restored images has been improved. Besides these, the existence of the minimizer of the Rician denoising model have also been established in this paper.

In this work, we propose a second-order version and a fourth-order version of a Cartesian grid-based boundary integral method for an interface problem of the Laplace equation on closely packed cells. When the cells are closely packed, the boundary integrals involved in the boundary integral formulation for the interface problem become nearly singular. Direct evaluation of the boundary integrals has accuracy issues. The grid-based method evaluates a boundary integral by first solving an equivalent, simple interface problem on a Cartesian grid with a fast Fourier transform based Poisson solver, then interpolating the grid solution to get values of the boundary integral at discretization points of the interface. The grid-based method presents itself as an alternative but accurate numerical method for evaluating nearly singular, singular and hyper-singular boundary integrals. This work can be regarded as a further development of the kernel-free boundary integral method [W.-J. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, Journal of Computational Physics, Vol. 227 (2007), pp. 1046-1074] for problems in unbounded domains. Numerical examples with both second-order and fourth-order versions of the grid-based method are presented to demonstrate accuracy of the method.

It has been shown in existing analysis that the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation is super-convergent when applied to the initial value problem of ordinary differential equations (ODEs) in that the discretization error is order 2s when s Gaussian nodes are used. Additionally, the discretized system can be solved accurately and efficiently using the spectral deferred correction (SDC) or Krylov deferred correction (KDC) method. In this paper, we combine the GRK formulation with the Method of Lines Transpose (MoL^T) to solve time-dependent parabolic partial differential equations (PDEs). For the GRK-MoL^T formulation, we show how the coupled spatial differential equations can be decoupled and efficiently solved using the SDC or KDC method. Preliminary analysis of the GRK-MoL^T algorithm reveals that the super-convergent property of the GRK formulation no longer holds in the PDE case for general boundary conditions, and there exists a new type of "stiffness" in the semi-discrete spatial elliptic differential equations. We present numerical experiments to validate the theoretical findings.

We develop a fast meshless algorithm for electrostatic interaction in an irregular domain with given potential boundary conditions, which is of importance in many applications such as electrochemical energy and electric structure calculations. The algorithm is based on an approximation of the Green's function using two-level image charges, in which the inner-layer charges are located nearby the boundary to eliminate the singularity of the induced polarization potential, and the outer-layer charges with fixed positions approximate the long-range tail of the potential. We find the number of inner-layer image charges can be very small and thus the total complexity of the algorithm is less expensive and potentially suitable for use in particle simulations. The numerical results show the performance of the algorithm is attractive. We also use the algorithm to investigate the electrostatic interaction for particles in a cylindrical nanopore and show that the electrostatic interaction within the pore has an exponential decay.

In this work, we propose and compare four different strategies for simulating the fluid model of quasi-three-dimensional streamer propagation, consisting of Poisson's equation for the particle velocity and two continuity equations for particle transport in the cylindrical coordinate system with angular symmetry. Each strategy involves one method for solving Poisson's equation, a discontinuous Galerkin method for solving the continuity equations, and a total variation-diminishing RungeKutta method in temporal discretization. The numerical methods for Poisson's equation include discontinuous Galerkin methods, the mixed finite element method, and the least-squares finite element method. The numerical method for continuity equations is the Oden-Babuška-Baumann discontinuous Galerkin method. A slope limiter for the DG methods in the cylindrical coordinate system is proposed to conserve the physical property. Tests and comparisons show that all four strategies are compatible in the sense that solutions to particle densities converge. Finally, different types of streamer propagation phenomena were simulated using the proposed method, including double-headed streamer in nitrogen and SF₆ between parallel plates, a streamer discharge in a point-to-plane gap.