124 research outputs found
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
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
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
We propose and rigourously analyze a multiscale time integrator Fourier
pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless
parameter which is inversely proportional to the speed of
light. In the nonrelativistic limit regime, i.e. , the
solution exhibits highly oscillatory propagating waves with wavelength
and in time and space, respectively. Due to the rapid
temporal oscillation, it is quite challenging in designing and analyzing
numerical methods with uniform error bounds in . We
present the MTI-FP method based on properly adopting a multiscale decomposition
of the solution of the Dirac equation and applying the exponential wave
integrator with appropriate numerical quadratures. By a careful study of the
error propagation and using the energy method, we establish two independent
error estimates via two different mathematical approaches as
and ,
where is the mesh size, is the time step and depends on the
regularity of the solution. These two error bounds immediately imply that the
MTI-FP method converges uniformly and optimally in space with exponential
convergence rate if the solution is smooth, and uniformly in time with linear
convergence rate at for all and optimally with
quadratic convergence rate at in the regimes when either
or . Numerical results are
reported to demonstrate that our error estimates are optimal and sharp.
Finally, the MTI-FP method is applied to study numerically the convergence
rates of the solution of the Dirac equation to those of its limiting models
when .Comment: 25 pages, 1 figur
Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime
We analyze rigorously error estimates and compare numerically
spatial/temporal resolution of various numerical methods for the discretization
of the Dirac equation in the nonrelativistic limit regime, involving a small
dimensionless parameter which is inversely proportional to
the speed of light. In this limit regime, the solution is highly oscillatory in
time, i.e. there are propagating waves with wavelength and
in time and space, respectively. We begin with several frequently used
finite difference time domain (FDTD) methods and obtain rigorously their error
estimates in the nonrelativistic limit regime by paying particular attention to
how error bounds depend explicitly on mesh size and time step as
well as the small parameter . Based on the error bounds, in order
to obtain `correct' numerical solutions in the nonrelativistic limit regime,
i.e. , the FDTD methods share the same
-scalability on time step: . Then we
propose and analyze two numerical methods for the discretization of the Dirac
equation by using the Fourier spectral discretization for spatial derivatives
combined with the exponential wave integrator and time-splitting technique for
temporal derivatives, respectively. Rigorous error bounds for the two numerical
methods show that their -scalability on time step is improved to
when . Extensive numerical results
are reported to support our error estimates.Comment: 34 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1511.0119
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
Numerical studies on quantized vortex dynamics in superfludity and superconductivity
Ph.DDOCTOR OF PHILOSOPH
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
Bootstrapping octagons in reduced kinematics from cluster algebras
Multi-loop scattering amplitudes/null polygonal Wilson loops in super-Yang-Mills are known to simplify significantly in reduced
kinematics, where external legs/edges lie in an dimensional subspace of
Minkowski spacetime (or boundary of the subspace). Since the edges
of a -gon with even and odd labels go along two different null directions,
the kinematics is reduced to two copies of . In the
simplest octagon case, we conjecture that all loop amplitudes and Feynman
integrals are given in terms of two overlapping functions (a special case
of two-dimensional harmonic polylogarithms): in addition to the letters of , there are two letters mixing
the two sectors but they never appear together in the same term; these are the
reduced version of four-mass-box algebraic letters. Evidence supporting our
conjecture includes all known octagon amplitudes as well as new computations of
multi-loop integrals in reduced kinematics. By leveraging this alphabet and
conditions on first and last entries, we initiate a bootstrap program in
reduced kinematics: within the remarkably simple space of overlapping
functions, we easily obtain octagon amplitudes up to two-loop NMHV and
three-loop MHV. We also briefly comment on the generalization to -gons in
terms of functions and beyond.Comment: 26 pages, several figures and tables, an ancilary fil
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