14,612 research outputs found
Particle based gPC methods for mean-field models of swarming with uncertainty
In this work we focus on the construction of numerical schemes for the
approximation of stochastic mean--field equations which preserve the
nonnegativity of the solution. The method here developed makes use of a
mean-field Monte Carlo method in the physical variables combined with a
generalized Polynomial Chaos (gPC) expansion in the random space. In contrast
to a direct application of stochastic-Galerkin methods, which are highly
accurate but lead to the loss of positivity, the proposed schemes are capable
to achieve high accuracy in the random space without loosing nonnegativity of
the solution. Several applications of the schemes to mean-field models of
collective behavior are reported.Comment: Communications in Computational Physics, to appea
Zoology of a non-local cross-diffusion model for two species
We study a non-local two species cross-interaction model with
cross-diffusion. We propose a positivity preserving finite volume scheme based
on the numerical method introduced in Ref. [15] and explore this new model
numerically in terms of its long-time behaviours. Using the so gained insights,
we compute analytical stationary states and travelling pulse solutions for a
particular model in the case of attractive-attractive/attractive-repulsive
cross-interactions. We show that, as the strength of the cross-diffusivity
decreases, there is a transition from adjacent solutions to completely
segregated densities, and we compute the threshold analytically for
attractive-repulsive cross-interactions. Other bifurcating stationary states
with various coexistence components of the support are analysed in the
attractive-attractive case. We find a strong agreement between the numerically
and the analytically computed steady states in these particular cases, whose
main qualitative features are also present for more general potentials
Spatial Coherence Resonance near Pattern-Forming Instabilities
The analogue of temporal coherence resonance for spatial degrees of freedom
is reported. Specifically, we show that spatiotemporal noise is able to
optimally extract an intrinsic spatial scale in nonlinear media close to (but
before) a pattern-forming instability. This effect is observed in a model of
pattern-forming chemical reaction and in the Swift-Hohenberg model of fluid
convection. In the latter case, the phenomenon is described analytically via an
approximate approach.Comment: 4 pages, 4 figure
Local well-posedness of the generalized Cucker-Smale model
In this paper, we study the local well-posedness of two types of generalized
Cucker-Smale (in short C-S) flocking models. We consider two different
communication weights, singular and regular ones, with nonlinear coupling
velocities for . For the singular
communication weight, we choose with and in dimension . For the regular case, we
select belonging to (L_{loc}^\infty \cap
\mbox{Lip}_{loc})(\mathbb{R}^d) and . We also
remark the various dynamics of C-S particle system for these communication
weights when
Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure
We analyse the asymptotic behaviour of solutions to the one dimensional
fractional version of the porous medium equation introduced by Caffarelli and
V\'azquez, where the pressure is obtained as a Riesz potential associated to
the density. We take advantage of the displacement convexity of the Riesz
potential in one dimension to show a functional inequality involving the
entropy, entropy dissipation, and the Euclidean transport distance. An argument
by approximation shows that this functional inequality is enough to deduce the
exponential convergence of solutions in self-similar variables to the unique
steady states
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