10 research outputs found
On a bifurcation value related to quasi-linear Schrodinger equations
By virtue of numerical arguments we study a bifurcation phenomenon occurring
for a class of minimization problems associated with the quasi-linear
Schrodinger equation.Comment: 9 page
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 blowup 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 adaption 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
High-order time-splitting Hermite and Fourier spectral methods
In this paper, we are concerned with the numerical solution of the time-dependent Gross-Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions: on the other hand. restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration
Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation
We present an effective adaptive procedure for the numerical approximation of
the steady-state Gross-Pitaevskii equation. Our approach is solely based on
energy minimization, and consists of a combination of gradient flow iterations
and adaptive finite element mesh refinements. Numerical tests show that this
strategy is able to provide highly accurate results, with optimal convergence
rates with respect to the number of freedom
Two-Level discretization techniques for ground state computations of Bose-Einstein condensates
This work presents a new methodology for computing ground states of
Bose-Einstein condensates based on finite element discretizations on two
different scales of numerical resolution. In a pre-processing step, a
low-dimensional (coarse) generalized finite element space is constructed. It is
based on a local orthogonal decomposition and exhibits high approximation
properties. The non-linear eigenvalue problem that characterizes the ground
state is solved by some suitable iterative solver exclusively in this
low-dimensional space, without loss of accuracy when compared with the solution
of the full fine scale problem. The pre-processing step is independent of the
types and numbers of bosons. A post-processing step further improves the
accuracy of the method. We present rigorous a priori error estimates that
predict convergence rates H^3 for the ground state eigenfunction and H^4 for
the corresponding eigenvalue without pre-asymptotic effects; H being the coarse
scale discretization parameter. Numerical experiments indicate that these high
rates may still be pessimistic.Comment: Accepted for publication in SIAM J. Numer. Anal., 201
Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii
eigenvalue problem based on an energy inner product that depends on time
through the density of the flow itself. The gradient flow is well-defined and
converges to an eigenfunction. For ground states we can quantify the
convergence speed as exponentially fast where the rate depends on spectral gaps
of a linearized operator. The forward Euler time discretization of the flow
yields a numerical method which generalizes the inverse iteration for the
nonlinear eigenvalue problem. For sufficiently small time steps, the method
reduces the energy in every step and converges globally in to an
eigenfunction. In particular, for any nonnegative starting value, the ground
state is obtained. A series of numerical experiments demonstrates the
computational efficiency of the method and its competitiveness with established
discretizations arising from other gradient flows for this problem
BEC2HPC: a HPC spectral solver for nonlinear Schrödinger and Gross-Pitaevskii equations. Stationary states computation
International audienceWe present BEC2HPC which is a parallel HPC spectral solver for computing the ground states of the nonlinear Schrödinger equation and the Gross-Pitaevskii equation (GPE) modeling rotating Bose-Einstein condensates (BEC). Considering a standard pseudo-spectral discretization based on Fast Fourier Transforms (FFTs), the method consists in finding the numerical solution of the energy functional minimization problem under normalization constraint by using a preconditioned nonlinear conjugate gradient method. We present some numerical simulations and scalability results for the 2D and 3D problems to obtain the stationary states of BEC with fast rotation and large nonlinearities. The code takes advantage of existing HPC libraries and can itself be leveraged to implement other numerical methods like e.g. for the dynamics of BECs
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