13,649 research outputs found
Equivalent partial differential equations of a lattice Boltzmann scheme
AbstractWe show that when we formulate the lattice Boltzmann equation with a small-time step Δt and an associated space scale Δx, a Taylor expansion joined with the so-called equivalent equation methodology leads to establishing macroscopic fluid equations as a formal limit. We recover the Euler equations of gas dynamics at the first order and the compressible Navier–Stokes equations at the second order
General fourth order Chapman-Enskog expansion of lattice Boltzmann schemes
In order to derive the equivalent partial differential equations of a lattice
Boltzmann scheme,the Chapman Enskog expansion is very popular in the lattive
Boltzmann community. A maindrawback of this approach is the fact that
multiscale expansions are used without any clearmathematical signification of
the various variables and operators. Independently of thisframework, the Taylor
expansion method allows to obtain formally the equivalent partialdifferential
equations. In this contribution, we prove that both approaches give
identicalresults with acoustic scaling for a very general family of lattice
Boltzmann schemes and upto fourth order accuracy. Examples with a single scalar
conservation illustrate our purpose
Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme
Using the Taylor expansion method, we show that it is possible to improve the
lattice Boltzmann method for acoustic applications. We derive a formal
expansion of the eigenvalues of the discrete approximation and fit the
parameters of the scheme to enforce fourth order accuracy. The corresponding
discrete equations are solved with the help of symbolic manipulation. The
solutions are explicited in the case of D3Q27 lattice Boltzmann scheme. Various
numerical tests support the coherence of this approach.Comment: 23 page
Lattice Boltzmann schemes with relative velocities
In this contribution, a new class of lattice Boltzmann schemes is introduced
and studied. These schemes are presented in a framework that generalizes the
multiple relaxation times method of d'Humi\`eres. They extend also the Geier's
cascaded method. The relaxation phase takes place in a moving frame involving a
set of moments depending on a given relative velocity field. We establish with
the Taylor expansion method that the equivalent partial differential equations
are identical to the ones obtained with the multiple relaxation times method up
to the second order accuracy. The method is then performed to derive the
equivalent equations up to third order accuracy
Moving charged particles in lattice Boltzmann-based electrokinetics
The motion of ionic solutes and charged particles under the influence of an
electric field and the ensuing hydrodynamic flow of the underlying solvent is
ubiquitous in aqueous colloidal suspensions. The physics of such systems is
described by a coupled set of differential equations, along with boundary
conditions, collectively referred to as the electrokinetic equations. Capuani
et al. [J. Chem. Phys. 121, 973 (2004)] introduced a lattice-based method for
solving this system of equations, which builds upon the lattice Boltzmann
algorithm for the simulation of hydrodynamic flow and exploits computational
locality. However, thus far, a description of how to incorporate moving
boundary conditions into the Capuani scheme has been lacking. Moving boundary
conditions are needed to simulate multiple arbitrarily-moving colloids. In this
paper, we detail how to introduce such a particle coupling scheme, based on an
analogue to the moving boundary method for the pure LB solver. The key
ingredients in our method are mass and charge conservation for the solute
species and a partial-volume smoothing of the solute fluxes to minimize
discretization artifacts. We demonstrate our algorithm's effectiveness by
simulating the electrophoresis of charged spheres in an external field; for a
single sphere we compare to the equivalent electro-osmotic (co-moving) problem.
Our method's efficiency and ease of implementation should prove beneficial to
future simulations of the dynamics in a wide range of complex nanoscopic and
colloidal systems that was previously inaccessible to lattice-based continuum
algorithms
Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling
We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the
simulation of diffusive processes. When the mesh is refined while holding the
diffusivity constant, we first obtain asymptotic convergence. When the mesh
size tends to zero, however, this convergence breaks down in a curious fashion,
and we observe qualitative discrepancies from analytical solutions of the heat
equation. In this work, a new asymptotic analysis is derived to explain this
phenomenon using the Taylor expansion method, and a partial differential
equation of acoustic type is obtained in the asymptotic limit. We show that the
error between the D1Q3 numerical solution and a finite-difference approximation
of this acoustic-type partial differential equation tends to zero in the
asymptotic limit. In addition, a wave vector analysis of this asymptotic regime
demonstrates that the dispersion equation has nontrivial complex eigenvalues, a
sign of underlying propagation phenomena, and a portent of the unusual
convergence properties mentioned above
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