649 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
Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime
We present several numerical methods and establish their error estimates for
the discretization of the nonlinear 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 the conservative Crank-Nicolson finite difference (CNFD) method and
establish rigorously its error estimate which depends explicitly on the mesh
size and time step as well as the small parameter . Based on the error bound, in order to obtain `correct' numerical solutions
in the nonrelativistic limit regime, i.e. , the CNFD method
requests the -scalability: and
. Then we propose and analyze two numerical methods
for the discretization of the nonlinear 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 is improved to and
when compared with the CNFD method. Extensive
numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with
arXiv:1504.0288
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
Research on the Current Situation and Guiding Path of College Students’ Participation in Charitable Activities
As the traditional virtue of the Chinese nation, the excellent quality of helping others is an important support for building a harmonious socialist society. Charity activities are an important form to carry forward its excellent quality. College students as the future hope of the motherland, by encouraging and guiding college students to participate in charitable activities to cultivate relevant excellent quality is the important content of shaping contemporary higher education talents and realizing their personal life value, but at present, there are still problems in college students’ participation in charitable activities that need to be solved urgently. Taking YuLin University as an example, this paper points out the relevant theoretical concepts, analyzes the significance of college students’ participation in public welfare and charitable activities. Through data investigation, it is concluded that there are some problems in YuLin University students’ participation in public welfare and charitable activities, such as less participation, impure participation motivation, single participation information acquisition channel and distrust of relevant groups. This paper puts forward the improvement path from the aspects of individual, family, school and society in order to play an optimization role
- …