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

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