Conference Agenda

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Session Overview
Method and Analysis II
Thursday, 30/June/2022:
10:50am - 11:50am

Session Chair: Li-Shi Luo, CSRC
Location: Michel Crépeau's Lecture Hall, Pôle Communication, La Rochelle University

Pôle Communication Multimédia Reseaux, La Rochelle University, 44 Avenue Albert Einstein, La Rochelle.

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10:50am - 11:10am

A consistent discrete divergence-free condition in lattice Boltzmann magnetohydrodynamics: a data-driven approach

Dellar, Paul

University of Oxford, United Kingdom

Magnetohydrodynamics combines the Navier-Stokes and Maxwell equations to describe the flow of electrically conducting fluids in magnetic fields. Maxwell’s equations require the magnetic field to evolve in a way that keeps the divergence of the magnetic field zero. This is analogous to the incompressibility condition for a fluid, except there is no analogue of a pressure in Maxwell’s equations.

Lattice Boltzmann magnetohydrodynamics relies upon kinetic representations of both the fluid and the electromagnetic field. There is an extra kinetic degree of freedom that represents the divergence of the magnetic field for slowly varying solutions, but its relation to any discrete approximation to the divergence of the magnetic field on the lattice has previously been mysterious.

We show empirically, using data from simulations, that there is an optimal finite difference stencil for approximating the discrete divergence of the magnetic field, with parameters that depend on the relaxation times for the different components of the kinetic representation of the electromagnetic field. We then show that the parameters for the optimal stencil can be derived analytically using operator algebra techniques, and that they resemble formulas for the optimal placement of no-slip boundaries between lattice points.

Finally, we show that adjusting the relaxation time for the kinetic degree of freedom mentioned above implements an extended magnetohydrodynamics with an extra scalar field to maintain the divergence-free condition. The joint evolution of this kinetic degree of freedom and the optimal finite difference approximation for the discrete divergence of the magnetic field matches analytical solutions of the extended magnetohydrodynamics equations.

11:10am - 11:30am

Conservative models for the compressible hybrid lattice Boltzmann method

Wissocq, Gauthier; Coratger, Thomas; Farag, Gabriel; Zhao, Song; Boivin, Pierre; Sagaut, Pierre

Aix-Marseille Université, France

A new methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum and energy equations are solved by different methods. It is here designed for developing new numerical discretizations of the total energy equation, adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is written under a finite-volume formulation involving fluxes, which are included in the total energy equation as numerical corrections. Three models are subsequently derived: a first-order upwind, a Lax-Wendroff and a MUSCL-Hancock schemes. They are assessed on standard academic test cases for compressible flows, with and without discontinuitities. Three key features are exhibited: 1) the models are conservative by construction, recovering correct jump relations across shock waves, 2) the stability and accuracy of entropy modes can be explicitly controlled, 3) the low dissipation of the LBM for isentropic phenomena is preserved.

11:30am - 11:50am

Limit consistency of lattice Boltzmann equations

Simonis, Stephan; Krause, Mathias J.

Karlsruhe Institute of Technology (KIT), Germany

We establish the notion of limit consistency as a novel technique to formally prove the consistency of lattice Boltzmann equations (LBE) to a given partial differential equation (PDE). For the purpose of illustration, the incompressible Navier–Stokes equations (NSE) are used as a paragon. Based upon the proven diffusion limit [L. Saint-Raymond (2003), doi: 10.1016/S0012-9593(03)00010-7] of the BGK Boltzmann equation (BGKBE) towards the NSE, we provide a successive discretization by nesting conventional Taylor expansions and finite differences. Tracking the discretization state of the domain for the particle distribution functions, we measure truncation errors at all levels within the derivation procedure. Via parametrizing equations and proving limit consistency of the resulting sequences, we retain the path towards the targeted PDE at each step of discretization, i.e. for the discrete velocity BGKBE (DVBGKBE) and the space-time discretized lattice BGKBE (LBGKBE). As a direct result, we unfold the discretization technique of lattice Boltzmann methods as chaining finite differences and provide a generic top-down derivation of the numerical scheme.

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