In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical implementation, as well as showing a few practical examples demonstrating its capabilities.[-]

(Work in collaboration with C. Bardos and I. Moyano). Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $\sigma \to +\infty$ and $1 − \alpha ∼ C/\sigma$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of
a kinetic model which is based on the harmonic extension definition of $\sqrt{−\Delta}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (“heavy tails”) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273–280].[-]

We develop a numerical method for the Levy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold non locality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other comes from long-time/small mean-free-path scaling, which calls for a uniform stable solver. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply Chebyshev polynomial based pseudo-spectral method. To resolve the second issue, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically.[-]

What would be the impact of an environment change on the persistence and the genetic/phenotypic distribution of a population? We present some integro-differential models describing the evolutionary adaptation of asexual phenotypically structured populations subject to mutation and selection in changing environments. Using an approach based on Hamilton-Jacobi equations, we provide an asymptotic analysis of such equations in the regime of small mutational variance. This analysis allows us to characterize different evolutionary outcomes depending on the type of the environmental change.[-]

What would be the impact of an environment change on the persistence and the genetic/phenotypic distribution of a population? We present some integro-differential models describing the evolutionary adaptation of asexual phenotypically structured populations subject to mutation and selection in changing environments. Using an approach based on Hamilton-Jacobi equations, we provide an asymptotic analysis of such equations in the regime of small mutational variance. This analysis allows us to characterize different evolutionary outcomes depending on the type of the environmental change.[-]