Section: New Results

Asymptotic analysis

In [33], C. Cancès and co-workers derive the porous medium equation as the hydrodynamic limit of an interacting particle system which belongs to the family of exclusion processes with nearest neighbor exchanges. The main outcome of this work is to allow regions with vanishing density, where the dynamics turns out to degenerate. The convergence builds on a generalization of the entropy method and on suitable regularization of the dynamics.

In [29], A. Ait Hammou Oulhaj, C. Cancès, C. Chainais-Hillairet et al. study analytically and numerically the large time behavior of the solutions to a two-phase extension of the porous medium equation, which models the so-called seawater intrusion problem. They identify the self-similar solutions that correspond to steady states of a rescaled version of the problem. They finally provide numerical illustrations of the stationary states and exhibit numerical convergence rates.

In [13], C. Chainais-Hillairet et al. propose a new proof of existence of a solution to the scheme introduced in [63] which does not require any assumption on the time step. The result relies on the application of a topological degree argument which is based on the positivity and on uniform-in-time upper bounds of the approximate densities. They also establish uniform-in-time lower bounds satisfied by the approximate densities. These uniform-in-time upper and lower bounds ensure the exponential decay of the scheme towards the thermal equilibrium as shown in [63].

In [38], C. Chainais-Hillairet and M. Herda study the large-time behavior of solutions to Finite Volume discretizations of convection-diffusion equations or systems endowed with non-homogeneous Dirichlet and Neumann type boundary conditions. Their results concern various linear and nonlinear models such as Fokker–Planck equations, porous media equations, or drift-diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. They show that in the framework of Finite Volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserve this exponential decay of the discrete solution to the discrete steady state of the scheme.

In [32], M. Herda, T. Rey et al. are interested in the asymptotic analysis of a Finite Volume scheme for one-dimensional linear kinetic equations, with either Fokker–Planck or linearized BGK collision operator. Thanks to appropriate uniform estimates, they establish that the proposed scheme is asymptotic-preserving in the diffusive limit. Moreover, they adapt to the discrete framework the hypocoercivity method proposed by [80] to prove the exponential return to equilibrium of the approximate solution. They obtain decay estimates that are uniform in the diffusive limit. Finally, they present an efficient implementation of the proposed numerical schemes, and perform numerous numerical simulations assessing their accuracy and efficiency in capturing the correct asymptotic behaviors of the models.

In [26], M. Herda et al. consider various sets of Vlasov–Fokker–Planck equations modeling the dynamics of charged particles in a plasma under the effect of a strong magnetic field. For each of them, in a regime where the strength of the magnetic field is effectively stronger than that of collisions, they first formally derive asymptotically reduced models. In this regime, strong anisotropic phenomena occur; while equilibrium along magnetic field lines is asymptotically reached the asymptotic models capture a nontrivial dynamics in the perpendicular directions. They do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self-consistent electric fields are neglected they provide a rigorous mathematical justification of the formally derived systems. In this last step they provide a complete control on solutions by developing anisotropic hypocoercive estimates.