On Triangular Lattice Boltzmann Schemes for Scalar Problems

Abstract

We propose to extend the d'Humières 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.

Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

“…They are denominated (in  this order) by ρ, J x , J y , ε, x x , x y , q x , q y , h. The lines of this invertible matrix are chosen orthogonal. For isothermal flows, the conserved moments (23) W = ρ , J x , J y t correspond to the three first lines of the d'Humières matrix (22). The non-conserved moments complete the family:…”

“…In the relation ( 25), we have put in evidence the block decomposition (8) for the D2Q9 scheme with the conserved moments precised in (23). Even if the space differential operators commute, the matrices that compose the momentum-velocity operator matrix does not commute!…”

“…Observe that the main result is implemented with this framework in the "pyLBM" ' software [30]. The next step is the introduction of source terms, in the spirit proposed in [25], and other geometries like triangles, in a way suggested in [23]. The adaptation of the Taylor expansion method to other variants of lattice Boltzmann schemes as recentered schemes …”

Unexpected convergence of lattice Boltzmann schemes

“…It is well known [17] that the time step must be proportional to the square of the spacial step in order for the method to be stable. An asymptotic analysis can be done for this simple lattice Boltzmann thermic model, as, e.g., in our contribution [7], and we obtain again the heat equation (9) as the scaling limit of the model. With this diffusive scaling, the parameters σ and s J remain constant if the thermal diffusivity is given and the mesh size ∆x tends to zero.…”

“…When we use diffusive scaling, this dispersion equation can be adapted in order to recover the heat equation at zero order of accuracy. It is then equivalent to the Taylor expansion method with the diffusive scaling, as used in [7].…”