On triangular lattice Boltzmann schemes for scalar problems

TL;DR

This paper extends the lattice Boltzmann method to triangular meshes for scalar problems like heat transfer, demonstrating second-order convergence and opening questions about super-convergence potential.

Contribution

It introduces a novel extension of the lattice Boltzmann scheme to triangular meshes, enabling particle-based simulations without finite volume methods.

Findings

Achieved second-order convergence in heat equation simulations.

Validated the scheme's effectiveness on triangular lattices.

Raised questions about potential super-convergence phenomena.

Abstract

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.