289 research outputs found
On triangular lattice Boltzmann schemes for scalar problems
We propose to extend the d'Humi\'eres version of the lattice Boltzmann scheme
to triangular meshes. We use Bravais lattices or more general lattices with the
property that the degree of each internal vertex is supposed to be constant. On
such meshes, it is possible to define the lattice Boltzmann scheme as a
discrete particle method, without need of finite volume formulation or
Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat
equation and perform an asymptotic analysis with the Taylor expansion method
for two schemes named D2T4 and D2T7. The results show a convergence up to
second order accuracy and set new questions concerning a possible
super-convergence.Comment: 23 page
Lattice Boltzmann model approximated with finite difference expressions
We show that the asymptotic properties of the link-wise artificial
compressibility method are not compatible with a correct approximation of fluid
properties. We propose to adapt the previous method through a framework
suggested by the Taylor expansion method and to replace first order terms in
the expansion by appropriate three or five points finite differences and to add
non linear terms. The "FD-LBM" scheme obtained by this method is tested in two
dimensions for shear wave, Stokes modes and Poiseuille flow. The results are
compared with the usual lattice Boltzmann method in the framework of multiple
relaxation times
On lattice Boltzmann scheme, finite volumes and boundary conditions
We develop the idea that a natural link between Boltzmann schemes and finite
volumes exists naturally: the conserved mass and momentum during the collision
phase of the Boltzmann scheme induces general expressions for mass and momentum
fluxes. We treat a unidimensional case and focus our development in two
dimensions on possible flux boundary conditions. Several test cases show that a
high level of accuracy can be achieved with this scheme
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
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
Generalized bounce back boundary condition for the nine velocities two-dimensional lattice Boltzmann scheme
International audienceIn a previous work, we have proposed a method for the analysis of the bounce back boundary condition with the Taylor expansion method in the linear case. In this work two new schemes of modified bounce back are proposed. The first one is based on the expansion of the iteration of the internal scheme of the lattice Boltzmann method. The analysis puts in evidence some defects and a generalized version is proposed with a set of essentially four possible parameters to adjust. We propose to reduce this number to two with the elimination of spurious density first order terms. Thus a new scheme for bounce back is found exact up to second order and allows an accurate simulation of the Poiseuille flow for a specific combination of the relaxation and boundary coefficients. We have validated the general expansion of the value in the first cell in terms of given values on the boundary for a stationary ''accordion'' test case
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