50 research outputs found
A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme
We consider a mono-dimensional two-velocities scheme used to approximate the
solutions of a scalar hyperbolic conservative partial differential equation. We
prove the convergence of the discrete solution toward the unique entropy
solution by first estimating the supremum norm and the total variation of the
discrete solution, and second by constructing a discrete kinetic
entropy-entropy flux pair being given a continuous entropy-entropy flux pair of
the hyperbolic system. We finally illustrate our results with numerical
simulations of the advection equation and the Burgers equation
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
On the stability of a relative velocity lattice Boltzmann scheme for compressible Navier-Stokes equations
This paper studies the stability properties of a two dimensional relative
velocity scheme for the Navier-Stokes equations. This scheme inspired by the
cascaded scheme has the particularity to relax in a frame moving with a
velocity field function of space and time. Its stability is studied first in a
linear context then on the non linear test case of the Kelvin-Helmholtz
instability. The link with the choice of the moments is put in evidence. The
set of moments of the cascaded scheme improves the stability of the
d'Humi\`eres scheme for small viscosities. On the contrary, a relative velocity
scheme with the usual set of moments deteriorates the stability
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
Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9
In this paper, we recall the linear version of the lattice Boltzmann schemes
in the framework proposed by d'Humi\'eres. According to the equivalent
equations we introduce a definition for a scheme to be isotropic at some order.
This definition is chosen such that the equivalent equations are preserved by
orthogonal transformations of the frame. The property of isotropy can be read
through a group operation and then implies a sequence of relations on
relaxation times and equilibrium states that characterizes a lattice Boltzmann
scheme. We propose a method to select the parameters of the scheme according to
the desired order of isotropy. Applying it to the D2Q9 scheme yields the
classical constraints for the first and second orders and some non classical
for the third and fourth orders
A stability property for a mono-dimensional three velocities scheme with relative velocity
In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied