Impact of collision models on the physical properties and the stability of lattice Boltzmann methods

The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Advanced Diagnosis Techniques in Medical Imaging

The Boltzmann distribution was derived in this chapter. The Boltzmann equation was explained next to the main difficulty of this equation, the integral of the collision operator, which was solved by the BGK-approximation where a long-term substitute is essential. The discretization of the Boltzmann comparison with the BGK-approximation was introduced along with the lattice and the different lattice configurations to define the lattice framework where the method is applied. Also, in this framework, the algorithm of the process was described. The boundary conditions were summarised, where one can see that they represent macroscopic conditions acting locally in every node.

Matrix-valued quantum lattice Boltzmann method

We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.

CONVERGENCE OF A BGK APPROXIMATION OF THE ISENTROPIC EULER EQUATIONS

We consider a discrete kinetic approximation of the isentropic Euler equations, and establish the local convergence of the solutions of these relaxation systems to those of the hydrodynamic equations in the hyperbolic limit. We rely on modulated entropy methods and cover the time interval in which the latter admits smooth solutions.

A Bhatnagar–Gross–Krook approximation to scalar conservation laws with discontinuous flux

We study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.