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Construction and analysis of lattice Boltzmann methods applied to a 1D convection-diffusion equation. (English) Zbl 1305.76081

Summary: To solve the 1D (linear) convection-diffusion equation, we construct and we analyze two LBM schemes built on the D1Q2 lattice. 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 in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in \(L^\infty\) of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in \(L^\infty\) under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results.

MSC:

76M28 Particle methods and lattice-gas methods
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
65C05 Monte Carlo methods
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