542 research outputs found
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
The multi-configurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems
The evolution of Bose-Einstein condensates is amply described by the
time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to
reside in a single time-dependent one-particle state throughout the propagation
process. In this work, we go beyond mean-field and develop an essentially-exact
many-body theory for the propagation of the time-dependent Schr\"odinger
equation of interacting identical bosons. In our theory, the time-dependent
many-boson wavefunction is written as a sum of permanents assembled from
orthogonal one-particle functions, or orbitals, where {\it both} the expansion
coefficients {\it and} the permanents (orbitals) themselves are {\it
time-dependent} and fully determined according to a standard time-dependent
variational principle. By employing either the usual Lagrangian formulation or
the Dirac-Frenkel variational principle we arrive at two sets of coupled
equations-of-motion, one for the orbitals and one for the expansion
coefficients. The first set comprises of first-order differential equations in
time and non-linear integro-differential equations in position space, whereas
the second set consists of first-order differential equations with
time-dependent coefficients. We call our theory multi-configurational
time-dependent Hartree for bosons, or MCTDHB(), where specifies the
number of time-dependent orbitals used to construct the permanents. Numerical
implementation of the theory is reported and illustrative numerical examples of
many-body dynamics of trapped Bose-Einstein condensates are provided and
discussed.Comment: 30 pages, 2 figure
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
Generalization of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schr\"odinger type
The present work is concerned with the extension of modified potential
operator splitting methods to specific classes of nonlinear evolution
equations. The considered partial differential equations of Schr{\"o}dinger and
parabolic type comprise the Laplacian, a potential acting as multiplication
operator, and a cubic nonlinearity. Moreover, an invariance principle is
deduced that has a significant impact on the efficient realisation of the
resulting modified operator splitting methods for the Schr{\"o}dinger case.}
Numerical illustrations for the time-dependent Gross--Pitaevskii equation in
the physically most relevant case of three space dimensions and for its
parabolic counterpart related to ground state and excited state computations
confirm the benefits of the proposed fourth-order modified operator splitting
method in comparison with standard splitting methods.
The presented results are novel and of particular interest from both, a
theoretical perspective to inspire future investigations of modified operator
splitting methods for other classes of nonlinear evolution equations and a
practical perspective to advance the reliable and efficient simulation of
Gross--Pitaevskii systems in real and imaginary time.Comment: 30 pages, 6 figure
A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates
Numerical computations of stationary states of fast-rotating Bose-Einstein
condensates require high spatial resolution due to the presence of a large
number of quantized vortices. In this paper we propose a low-order finite
element method with mesh adaptivity by metric control, as an alternative
approach to the commonly used high order (finite difference or spectral)
approximation methods. The mesh adaptivity is used with two different numerical
algorithms to compute stationary vortex states: an imaginary time propagation
method and a Sobolev gradient descent method. We first address the basic issue
of the choice of the variable used to compute new metrics for the mesh
adaptivity and show that simultaneously refinement using the real and imaginary
part of the solution is successful. Mesh refinement using only the modulus of
the solution as adaptivity variable fails for complicated test cases. Then we
suggest an optimized algorithm for adapting the mesh during the evolution of
the solution towards the equilibrium state. Considerable computational time
saving is obtained compared to uniform mesh computations. The new method is
applied to compute difficult cases relevant for physical experiments (large
nonlinear interaction constant and high rotation rates).Comment: to appear in J. Computational Physic
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