Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

Stéphane Dellacherie (1)

1 CEA-DES (ex-DEN) - CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire)

We construct and we analyze two LBM schemes applied to a 1D convection-diffusion equation. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show that these LBM schemes are equivalent to a finite difference type scheme for different boundary conditions. This allows us, firstly, to prove the convergence in L^inf of these LBM schemes and to obtain discrete maximum principles in the case of the 1D heat equation for any Delta t of the order of Delta x^2. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. By proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D heat equation. At last, we present numerical applications justifying these results.

Keywords: Lattice Boltzmann method; Chapman-Enskog expansion; Hilbert expansion; convection-diffusion equation; heat equation; Du Fort-Frankel scheme; stability and convergence in L^inf; discrete maximum principle; Monte-Carlo algorithm.; Monte-Carlo algorithm

Domains: Numerical Analysis [math.NA]

Stéphane Dellacherie. Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation. 2012. ⟨hal-00717873v2⟩