Together with my collaborator Lorenzo Pareschi, we have just submitted a new preprint entitled Residual equilibrium schemes for time dependent partial differential equations. This work extends the note about steady state preserving spectral methods to a large class of dissipative PDE’s. Here is the abstract:
Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.
My collaborator Giuseppe Toscani and myself submitted a new paper, entitled Large-time behavior of the solutions to Rosenau type approximations to the heat equation.
We study in this paper the validity of the approximation to the linear diffusion equation proposed by Rosenau as a regularized version of the Chapman-Enskog expansion of hydrodynamics. This approximation essentially is realized by substituting the heat equation with a linear kinetic equation of Boltzmann type, describing collisions of particles with a fixed background. This remark allows to consider the Rosenau approximation as a particular realization of a model Boltzmann equation, in which the background distribution is a general probability density with bounded variance. In addition to the Rosenau distribution, we considered also a point masses background, which furnishes the central difference scheme to solve numerically the linear diffusion equation.
I finished a new paper with my collaborator (and advisor) Francis Filbet: A Rescaling Velocity Method for Dissipative Kinetic Equations – Applications to Granular Media.
We present in this paper a new numerical algorithm based on a relative energy scaling for collisional kinetic equations allowing to study numerically their long time behavior, without the usual problems related to the change of scales in velocity variables. Several applications are presented for Boltzmann like equations. This method is particularly efficient for numerical simulations of the granular gases equation with dissipative energy: it allows to study accurately the long time behavior of this equation and is very well suited for the study of clustering phenomena.