Three-dimensional lattice Boltzmann model for electrodynamics

Abstract

In this paper we introduce a three-dimensional Lattice-Boltzmann model that recovers in the continuous limit the Maxwell equations in materials. In order to build conservation equations with antisymmetric tensors, like the Faraday law, the model assigns four auxiliary vectors to each velocity vector. These auxiliary vectors, when combined with the distribution functions, give the electromagnetic fields. The evolution is driven by the usual Bhatnager-Gross-Krook (BGK) collision rule, but with a different form for the equilibrium distribution functions. This lattice Bhatnager-Gross-Krook (LBGK) model allows us to consider for both dielectrics and conductors with realistic parameters, and therefore it is adequate to simulate the most diverse electromagnetic problems, like the propagation of electromagnetic waves (both in dielectric media and in waveguides), the skin effect, the radiation pattern of a small dipole antenna and the natural frequencies of a resonant cavity, all with 2% accuracy. Actually, it shows to be one order of magnitude faster than the original Finite-difference time-domain (FDTD) formulation by Yee to reach the same accuracy. It is, therefore, a valuable alternative to simulate electromagnetic fields and opens lattice Boltzmann for a broad spectrum of new applications in electrodynamics.

Lattice Boltzmann model for resistive relativistic magnetohydrodynamics

“…This means that there are 13 different electric vectors and 7 different magnetic vectors. In order to solve the Maxwell equations by the LB model we can write Ampere's law (Faraday's law) as time derivative of electric (magnetic) field plus the divergence of an antisymmetric tensor [24]. We also consider the term(−µ 0 J) in Ampere's law (right hand side of Eq.…”

“…For the electromagnetic part, i.e., solving the Maxwell equations, the LB model for electrodynamics proposed in Ref. [24] is modified and extended for coupling with the fluid equations and to include the relativistic Ohm's law. It should be mentioned that in the non-relativistic context, there are several LB models for resistive MHD [25][26][27], especially for simulating magnetic reconnection [28,29].…”

Lattice Boltzmann model for the simulation of the wave equation in curvilinear coordinates

“…Lattice-Boltzmann methods (LBM) were first introduced as mesoscopical models to simulate a wide variety of processes in fluid dynamics, from wind tunnels and turbulence to porous media and general rheology [2]; but later on they were extended to more general systems, like waves [3][4][5], electrodynamics [6], or even Quantum Mechanics [7] and, therefore, they can be considered nowdays as a general numerical scheme to solve differential equations that can be written as a set of conservation laws. In comparison with other numerical schemes (like finite-differences or finite-element methods) all variables needed to compute for the next time step at a single node are present in the node itself (before being moved to the neighbouring cells), making them perfect to run parallel on graphic cards.…”

“…In comparison with other numerical schemes (like finite-differences or finite-element methods) all variables needed to compute for the next time step at a single node are present in the node itself (before being moved to the neighbouring cells), making them perfect to run parallel on graphic cards. Additionally, LBM models like the one for electrodynamics, has shown to be up to five times faster than the mentioned methods even in serial calculations for the same precision order [6]. Because of these advantages, LBMs have gained the interest of a wide range of research areas and industrial applications; however, since most lattice-Boltzmann models assume a homogeneous and isotropic set of velocity vectors to move * amvelascos@unal.edu.co † jdmunozc@unal.edu.co the information from node to node, the computational domain has been restricted to a rectangular array of cubic cells, forcing the use of staircase approximations on curved boundaries and imposing three-dimensional simulation domains for systems that, because of axial or spherical symmetries, were essentially two-dimensional, with an exorbitant increase in computational costs.…”

Dean instability in double-curved channels

“…[7]. While it was originally designed to solve fluid flows, the LB method has even been applied to electrodynamics [8] and magnetohydrodynamics [9] as well as relativistic [10] and ultra-relativistic flows [11]. Most of the LB applications use standard Cartesian coordinates (e.g.…”