20 research outputs found
Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with
quadratic drift as a PDE model for the dynamics of bosons in the spatially
homogeneous setting. It is an open question whether this equation has solutions
exhibiting condensates in finite time. The main analytical challenge lies in
the continuation of exploding solutions beyond their first blow-up time while
having a linear diffusion term. We present a thoroughly validated time-implicit
numerical scheme capable of simulating solutions for arbitrarily large time,
and thus enabling a numerical study of the condensation process in the
Kaniadakis--Quarati model. We show strong numerical evidence that above the
critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model
in 3D form a condensate in finite time and converge in entropy to the unique
minimiser of the natural entropy functional at an exponential rate. Our
simulations further indicate that the spatial blow-up profile near the origin
follows a universal power law and that transient condensates can occur for
sufficiently concentrated initial data.Comment: To appear in Kinet. Relat. Model
Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation
We study a Lagrangian numerical scheme for solution of a nonlinear drift
diffusion equation on an interval. The discretization is based on the
equation's gradient flow structure with respect to the Wasserstein distance.
The scheme inherits various properties of the continuous flow, like entropy
monotonicity, mass preservation, metric contraction and minimum/maximum
principles. As the main result, we give a proof of convergence in the limit of
vanishing mesh size under a CFL-type condition. We also present results from
numerical experiments.Comment: 28 pages, 6 figure
Lecture Notes on Gradient Flows and Optimal Transport
We present a short overview on the strongest variational formulation for
gradient flows of geodesically -convex functionals in metric spaces,
with applications to diffusion equations in Wasserstein spaces of probability
measures. These notes are based on a series of lectures given by the second
author for the Summer School "Optimal transportation: Theory and applications"
in Grenoble during the week of June 22-26, 2009
Numerical simulation of continuity equations by evolving diffeomorphisms
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this properties with various examples in spatial dimension one and two