8,881 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
Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling
We consider a model system consisting of two reaction-diffusion equations,
where one species diffuses in a volume while the other species diffuses on the
surface which surrounds the volume. The two equations are coupled via a
nonlinear reversible Robin-type boundary condition for the volume species and a
matching reversible source term for the boundary species. As a consequence of
the coupling, the total mass of the two species is conserved. The considered
system is motivated for instance by models for asymmetric stem cell division.
Firstly we prove the existence of a unique weak solution via an iterative
method of converging upper and lower solutions to overcome the difficulties of
the nonlinear boundary terms. Secondly, our main result shows explicit
exponential convergence to equilibrium via an entropy method after deriving a
suitable entropy entropy-dissipation estimate for the considered nonlinear
volume-surface reaction-diffusion system.Comment: 31 page
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