23 research outputs found
Structure preserving schemes for mean-field equations of collective behavior
In this paper we consider the development of numerical schemes for mean-field
equations describing the collective behavior of a large group of interacting
agents. The schemes are based on a generalization of the classical Chang-Cooper
approach and are capable to preserve the main structural properties of the
systems, namely nonnegativity of the solution, physical conservation laws,
entropy dissipation and stationary solutions. In particular, the methods here
derived are second order accurate in transient regimes whereas they can reach
arbitrary accuracy asymptotically for large times. Several examples are
reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic
Problem
A Rosenau-type approach to the approximation of the linear Fokker--Planck equation
{The numerical approximation of the solution of the Fokker--Planck equation
is a challenging problem that has been extensively investigated starting from
the pioneering paper of Chang and Cooper in 1970. We revisit this problem at
the light of the approximation of the solution to the heat equation proposed by
Rosenau in 1992. Further, by means of the same idea, we address the problem of
a consistent approximation to higher-order linear diffusion equations
On a kinetic description of Lotka-Volterra dynamics
Owing to the analogies between the problem of wealth redistribution with
taxation in a multi-agent society, we introduce and discuss a kinetic model
describing the statistical distributions in time of the sizes of groups of
biological systems with prey-predator dynamic. While the evolution of the mean
values is shown to be driven by a classical Lotka-Volterra system of
differential equations, it is shown that the time evolution of the probability
distributions of the size of groups of the two interacting species is heavily
dependent both on a kinetic redistribution operator and the degree of
randomness present in the system. Numerical experiments are given to clarify
the time-behavior of the distributions of groups of the species
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
High order semi-implicit multistep methods for time dependent partial differential equations
We consider the construction of semi-implicit linear multistep methods which
can be applied to time dependent PDEs where the separation of scales in
additive form, typically used in implicit-explicit (IMEX) methods, is not
possible. As shown in Boscarino, Filbet and Russo (2016) for Runge-Kutta
methods, these semi-implicit techniques give a great flexibility, and allows,
in many cases, the construction of simple linearly implicit schemes with no
need of iterative solvers. In this work we develop a general setting for the
construction of high order semi-implicit linear multistep methods and analyze
their stability properties for a prototype linear advection-diffusion equation
and in the setting of strong stability preserving (SSP) methods. Our findings
are demonstrated on several examples, including nonlinear reaction-diffusion
and convection-diffusion problems
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
Hessian eigenvalue distribution in a random Gaussian landscape
The energy landscape of multiverse cosmology is often modeled by a
multi-dimensional random Gaussian potential. The physical predictions of such
models crucially depend on the eigenvalue distribution of the Hessian matrix at
potential minima. In particular, the stability of vacua and the dynamics of
slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues.
The Hessian eigenvalue distribution has been studied earlier, using the saddle
point approximation, in the leading order of expansion, where is the
dimensionality of the landscape. This approximation, however, is insufficient
for the small eigenvalue end of the spectrum, where sub-leading terms play a
significant role. We extend the saddle point method to account for the
sub-leading contributions. We also develop a new approach, where the eigenvalue
distribution is found as an equilibrium distribution at the endpoint of a
stochastic process (Dyson Brownian motion). The results of the two approaches
are consistent in cases where both methods are applicable. We discuss the
implications of our results for vacuum stability and slow-roll inflation in the
landscape.Comment: 33 pages, 10 figure
Structure preserving schemes for the continuum Kuramoto model: phase transitions
The construction of numerical schemes for the Kuramoto model is challenging
due to the structural properties of the system which are essential in order to
capture the correct physical behavior, like the description of stationary
states and phase transitions. Additional difficulties are represented by the
high dimensionality of the problem in presence of multiple frequencies. In this
paper, we develop numerical methods which are capable to preserve these
structural properties of the Kuramoto equation in the presence of diffusion and
to solve efficiently the multiple frequencies case. The novel schemes are then
used to numerically investigate the phase transitions in the case of identical
and non identical oscillators