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.
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.
I added a link to a paper I wrote with C. Mouhot and L. Pareschi: Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation. It deals with a class of algorithms for computing the Boltzmann collision operator for hard spheres, and therefore computing the solutions to the Boltzmann equation. This class of algorithms is known as discrete-velocity methods (DVM), and is highly stable, due to the nice positivity and conservation properties of the schemes. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N2d+1) where d is the dimension of the velocity space. In this paper, we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from O(N2d+1) to O(Nd log N), with almost no loss of accuracy.