566 research outputs found
Fourier-Splitting methods for the dynamics of rotating Bose-Einstein condensates
We present a new method to propagate rotating Bose-Einstein condensates
subject to explicitly time-dependent trapping potentials. Using algebraic
techniques, we combine Magnus expansions and splitting methods to yield any
order methods for the multivariate and nonautonomous quadratic part of the
Hamiltonian that can be computed using only Fourier transforms at the cost of
solving a small system of polynomial equations. The resulting scheme solves the
challenging component of the (nonlinear) Hamiltonian and can be combined with
optimized splitting methods to yield efficient algorithms for rotating
Bose-Einstein condensates. The method is particularly efficient for potentials
that can be regarded as perturbed rotating and trapped condensates, e.g., for
small nonlinearities, since it retains the near-integrable structure of the
problem. For large nonlinearities, the method remains highly efficient if
higher order p > 2 is sought. Furthermore, we show how it can adapted to the
presence of dissipation terms. Numerical examples illustrate the performance of
the scheme.Comment: 15 pages, 4 figures, as submitted to journa
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
Perfectly Matched Layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates
In this paper, we first propose a general strategy to implement the Perfectly Matched Layer (PML) approach in the most standard numerical schemes used for simulating the dynamics of nonlinear Schrödinger equations. The methods are based on the time-splitting [15] or relaxation [24] schemes in time, and finite element or FFT-based pseudospectral discretization methods in space. A thorough numerical study is developed for linear and nonlinear problems to understand how the PML approach behaves (absorbing function and tuning parameters) for a given scheme. The extension to the rotating Gross-Pitaevskii equation is then proposed by using the rotating Lagrangian coordinates transformation method [13, 16, 39], some numerical simulations illustrating the strength of the proposed approach
High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations
International audienceThe aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one-and two-components systems of Gross-Pitaevskii equations
Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques
We review phase space techniques based on the Wigner representation that
provide an approximate description of dilute ultra-cold Bose gases. In this
approach the quantum field evolution can be represented using equations of
motion of a similar form to the Gross-Pitaevskii equation but with stochastic
modifications that include quantum effects in a controlled degree of
approximation. These techniques provide a practical quantitative description of
both equilibrium and dynamical properties of Bose gas systems. We develop
versions of the formalism appropriate at zero temperature, where quantum
fluctuations can be important, and at finite temperature where thermal
fluctuations dominate. The numerical techniques necessary for implementing the
formalism are discussed in detail, together with methods for extracting
observables of interest. Numerous applications to a wide range of phenomena are
presented.Comment: 110 pages, 32 figures. Updated to address referee comments. To appear
in Advances in Physic
Newton-based alternating methods for the ground state of a class of multi-component Bose-Einstein condensates
The computation of the ground states of special multi-component Bose-Einstein
condensates (BECs) can be formulated as an energy functional minimization
problem with spherical constraints. It leads to a nonconvex quartic-quadratic
optimization problem after suitable discretizations. First, we generalize the
Newton-based methods for single-component BECs to the alternating minimization
scheme for multi-component BECs. Second, the global convergent alternating
Newton-Noda iteration (ANNI) is proposed. In particular, we prove the
positivity preserving property of ANNI under mild conditions. Finally, our
analysis is applied to a class of more general "multi-block" optimization
problems with spherical constraints. Numerical experiments are performed to
evaluate the performance of proposed methods for different multi-component
BECs, including pseudo spin-1/2, anti-ferromagnetic spin-1 and spin-2 BECs.
These results support our theory and demonstrate the efficiency of our
algorithms
Perfectly Matched Layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates
International audienceIn this paper, we first propose a general strategy to implement the Perfectly Matched Layer (PML) approach in the most standard numerical schemes used for simulating the dynamics of nonlinear Schrödinger equations. The methods are based on the time-splitting [15] or relaxation [24] schemes in time, and finite element or FFT-based pseudospectral discretization methods in space. A thorough numerical study is developed for linear and nonlinear problems to understand how the PML approach behaves (absorbing function and tuning parameters) for a given scheme. The extension to the rotating Gross-Pitaevskii equation is then proposed by using the rotating Lagrangian coordinates transformation method [13, 16, 39], some numerical simulations illustrating the strength of the proposed approach
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