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We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker–Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier–Stokes–Fokker–Planck system for dilute polymeric fluids. In this context the Fokker–Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for...
The dynamics of dendritic growth of a crystal in an undercooled melt is
determined by macroscopic diffusion-convection of heat and by capillary forces
acting on the nanometer scale of the solid-liquid interface width.
Its modelling is useful for instance in processing techniques based on casting.
The phase-field method is widely used to study evolution of such microstructural phase transformations on
a continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landau
equation...
We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown...
We consider multiscale systems for which only a fine-scale
model describing the evolution of individuals (atoms,
molecules, bacteria, agents) is given, while we are interested in the
evolution of the population density on coarse space and time
scales. Typically, this evolution is described by a coarse
Fokker-Planck equation.
In this paper, we consider a numerical procedure to compute the solution of
this Fokker-Planck equation directly on the coarse level, based on the
estimation of the unknown...
The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck...
We illustrate how some interesting new variational principles can be
used for the numerical approximation of solutions to certain (possibly
degenerate) parabolic partial differential equations. One remarkable
feature of the algorithms presented here is that derivatives do not
enter into the variational principles, so, for example, discontinuous
approximations may be used for approximating the heat equation. We
present formulae for computing a Wasserstein metric which enters
into the variational...
We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation . We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.
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