Around numerical analysis of lattice Boltzmann methods

Friday 2nd June 2023, Ecole polytechnique, France

Simulation of an air-conditioning system using pyLBM, see.

Purpose of the workshop

The aim of this workshop is to gather internationally recognized scientists from France and abroad—on the occasion of T. Bellotti's PhD defence—to discuss recent breakthroughs on lattice Boltzmann methods and related numerical analysis and schemes. These methods are widely praised for their efficiency and the large number of problems they are capable of tackling. Still, a large number of issues, especially of theoretical nature, remain open. For this reason, while also focusing on the applications of lattice Boltzmann schemes, the workshop pays particular attention to the theory behind these numerical methods.

Invited speakers

• Denise Aregba-Driollet, IMB - Université de Bordeaux, Discrete BGK models for the approximation of nonlinear conservative and nonconservative hyperbolic systems
• Paul J. Dellar, OCIAM - University of Oxford, Magic two-relaxation-time lattice Boltzmann schemes as macroscopic finite difference schemes for the Navier—Stokes and Maxwell equations
• Philippe Helluy, IRMA - Université de Strasbourg, Stable and accurate boundary conditions for kinetic approximations
• Li-Shi Luo, Old Dominion University, Title TBA
• Pierre Sagaut, Aix-Marseille Université, LBM as a general approach for solving conservation laws: recent results

The abstracts of these talks are provided at the bottom of the page.

Tentative programme

• 09:30-10:30: Talk by P. Helluy
• 10:30-11:00: Coffee break
• 11:00-12:00: Talk by P. Sagaut
• 12:00-12:30: Live demonstration of pyLBM by L. Gouarin
• 12:30-14:00: Lunch break
• 14:00-15:00: Talk by D. Aregba-Driollet
• 15:00-16:00: Talk by P. J. Dellar
• 16:00-16:30: Coffee break
• 16:30-17:30: Talk by L-S. Luo

Attending the workshop

People willing to attend the workshop are warmly welcomed free of charge. Still, we kindly ask you to register writing an email to thomas.bellotti[at]polytechnique.edu. We also plan to provide video-streaming for the workshop. If you want to receive the link, please contact us!

Date, venue and how to get there

The workshop is going to take place on Friday 2nd June 2023. The venue will be:

Conference Room "Amphithéâtre Pierre Faurre"

Ecole polytechnique, Route de Saclay, 91120 Palaiseau, France

How to get at Ecole polytechnique

How to get to the conference room

Organizers

T. Bellotti, L. Gouarin, B. Graille, M. Massot

Abstracts

Author:	Denise Aregba-Driollet
Title:	Discrete BGK models for the approximation of nonlinear conservative and nonconservative hyperbolic systems
Abstract:	Discrete BGK models are semilinear systems of transport equations with source terms which formally resemble the BGK model of the kinetic theory of gases. They provide a general relaxation approximation of quasilinear hyperbolic or hyperbolic-parabolic systems of PDEs. The framework for scalar conservation laws has been introduced in [R. Natalini, JDE 1998]. The generalization to systems of conservation laws and the design of related finite volume schemes was done in [D. Aregba-Driollet and R. Natalini, HYP98 Zurich proceedings, Birkhäuser, Basel, 1999]. Many papers were then written, both theoretically and numerically, in various physical and numerical contexts. In this talk, I shall give some results for conservative and nonconservative systems and try to link with the LBM.
Author:	Paul J. Dellar
Title:	Magic two-relaxation-time lattice Boltzmann schemes as macroscopic finite difference schemes for the Navier—Stokes and Maxwell equations
Abstract:	We interpret the lattice Boltzmann two-relaxation-time collision operator as prescribing different relaxation rates for the forward- and backward-propagating parts of each anti-parallel pair of discrete velocities. A particular “magic” combination sets the forward-propagating distributions to equilibrium. This allows us to construct closed finite difference schemes for the macroscopic variables alone across three time levels. We interpret these schemes as discretisations of first-order systems, the expected conservation law and a kinetic evolution equation the flux, and study these systems for the Navier—Stokes and Maxwell equations. In particular, we obtain a mimetic finite difference scheme for Maxwell’s equations that exactly preserves a discrete approximation to div B = 0, if it is satisfied initially, and otherwise evolves div B according to a telegraph equation. We obtain the magnetohydrodynamic induction equation solely from a non-relativistic approximation without needing a Chapman—Enskog expansion or equivalent slowly-varying approximation.
Author:	Philippe Helluy
Title:	Stable and accurate boundary conditions for kinetic approximations
Abstract:	The stability analysis of the lattice-Boltzmann method (LBM) is commonly examined using a Chapman-Enskog analysis. However, this approach may not always provide sufficient insights in certain cases. In this presentation, I will rather use an entropy stability analysis, initially proposed by Bouchut or Dubois, which has proven to be more reliable and effective in practical scenarios. I will demonstrate how this analysis can be applied to develop second-order stable boundary conditions for the LBM.
Author:	Pierre Sagaut
Title:	LBM as a general approach for solving conservation laws: recent results
Abstract:	Lattice Boltzmann Methods were first developed as an improvement of Lattice Gas Automata, and, in a second step, were re-interpreted as discretisations of the continuous Boltzmann equation and/or of the Discrete Velocity Boltzmann Equation. But, since a long time, LBM-type methods have been proposed to solve equations that completely escape the fluid dynamics framework (and therefore the Gas Kinetic Theory) such as solid mechanics, Maxwell equations or miscellaneous PDEs, rendering the classical physical interpretation of LBM as a discrete version of Gas Kinetic Theory meaningless. The talk will deal with recents results dealing with the analysis of LBM as a general tool to solve systems of PDEs that can be reformulated as a set of conservation laws, with some illustrations.

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