Residual equilibrium schemes for time dependent partial differential equations

The paper about Residual equilibrium schemes for time dependent partial differential equations written in collaboration with Lorenzo Pareschi has been accepted in Computers and Fluids. Yay!

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.

An efficient numerical method for solving the Boltzmann equation in multidimensions

Giacomo Dimarco, Raphaël Loubère, Jacek Narski and myself just submitted a preprint about An efficient numerical method for solving the Boltzmann equation in multidimensions. Yay!

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3D×3D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

Hydrodynamic limit of granular gases to pressureless Euler in dimension 1

Together with my collaborator Pierre-Emmanuel Jabin, we’ve just submitted a paper entitled Hydrodynamic limit of granular gases to pressureless Euler in dimension 1. This works aims to provide the first rigorous proof of the hydrodynamic limit of the kinetic, granular gases equation with strong inelasticity towards the fluid, pressureless Euler system, in dimension 1 of space and velocity. More precisely,

We investigate the behavior of granular gases in the limit of small Knudsen number, that is very frequent collisions. We deal with the strongly inelastic case, in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.

Residual equilibrium schemes for time dependent partial differential equations

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.

Towards an H-theorem for granular gases

Together with my collaborators M. I. García de Soria, P. Maynar, S. Mischler, C. Mouhot and E. Trizac, we submitted a new work entitled Towards an H-theorem for granular gases.

The H-theorem, originally derived at the level of Boltzmann non-linear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional H related to the original H, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic Direct Simulation Monte Carlo technique, and Molecular Dynamics) indicate that it is also non-increasing. Both driven and unforced cases are investigated.

An Exact Rescaling Velocity Method for some Kinetic Flocking Models

My collaborator Changhui Tan and myself just submitted a new research paper, entitled An Exact Rescaling Velocity Method for some Kinetic Flocking Models.

In this work, we discuss kinetic descriptions of flocking models, of the so-called Cucker-Smale and Motsch-Tadmor types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potential. We introduce a new exact rescaling velocity method, inspired by a recent work of Filbet and Rey, allowing to observe numerically the flocking behavior of the solutions to these equations, without a need of remeshing or taking a very fine grid in the velocity space. To stabilize the exact method, we also introduce a modification of the classical upwind finite volume scheme which preserves the physical properties of the solution, such as momentum conservation.