4,316 research outputs found
On the ground states and dynamics of space fractional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
In this paper, we propose some efficient and robust numerical methods to
compute the ground states and dynamics of Fractional Schr\"{o}dinger Equation
(FSE) with a rotation term and nonlocal nonlinear interactions. In particular,
a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal
interaction evaluation \cite{EMZ2015}. To compute the ground states, we
integrate the preconditioned Krylov subspace pseudo-spectral method \cite{AD1}
and the GauSum solver. For the dynamics simulation, using the rotating
Lagrangian coordinates transform \cite{BMTZ2013}, we first reformulate the FSE
into a new equation without rotation. Then, a time-splitting pseudo-spectral
scheme incorporated with the GauSum solver is proposed to simulate the new FSE
Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT
In this paper, we propose efficient and accurate numerical methods for
computing the ground state and dynamics of the dipolar Bose-Einstein
condensates utilising a newly developed dipole-dipole interaction (DDI) solver
that is implemented with the non-uniform fast Fourier transform (NUFFT)
algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation
(GPE) with a DDI term and present the corresponding two-dimensional (2D) model
under a strongly anisotropic confining potential. Different from existing
methods, the NUFFT based DDI solver removes the singularity by adopting the
spherical/polar coordinates in Fourier space in 3D/2D, respectively, thus it
can achieve spectral accuracy in space and simultaneously maintain high
efficiency by making full use of FFT and NUFFT whenever it is necessary and/or
needed. Then, we incorporate this solver into existing successful methods for
computing the ground state and dynamics of GPE with a DDI for dipolar BEC.
Extensive numerical comparisons with existing methods are carried out for
computing the DDI, ground states and dynamics of the dipolar BEC. Numerical
results show that our new methods outperform existing methods in terms of both
accuracy and efficiency.Comment: 26 pages, 5 figure
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT
International audienceWe present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schrödinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate NUFFT based method in [29] for nonlocal interactions where the singularity of the Fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates. We then extend the method to compute other nonlocal interactions whose Fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform. Many of these interactions do not decay at infinity in the physical space, which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain. The performance of our method against other existing methods is illustrated numerically, with particular attention on the effect of the size of the computational domain in the physical space. Finally, to study the ground state and dynamics of the NLSE, we propose efficient and accurate numerical methods by combining the NUFFT method for potential evaluation with the normalized gradient flow using backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively. Extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the NLSE with various nonlocal interactions. Numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency
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