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