The numerical solution of the transient heat conduction problem using the lattice Boltzmann method

Abstract

Abstract. The implementation of the lattice Boltzmann method (LBM) for the solution of the transient heat conduction problem is presented. The one dimensional task is considered and the different boundary conditions, specifically the Dirichlet, Neumann and Robin ones are taken into account. The D1Q2 lattice model is applied. To check the accuracy of the LBM algorithm, the same problems have been solved using the explicit variant of the finite difference method. In the final part of the paper, the results of computations are shown and the conclusions are formulated.

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

“…For example, (61) is equivalent to the Neumann boundary condition proposed in [6,39] (more precisely, (21) in [6] is equivalent to (59), and we easily show that (28) with q b = 0 in [6] is equivalent to (61) as soon as (61) is satisfied with n = 0). We can justify (62) -and, thus, (60) and (61) -with the following heuristic argument.…”

“…When u(x) = 0, the LBM scheme (35) is identical to the one proposed in [4,7,32,6] to solve the diffusion equation. When u(x) is a constant u 0 , (35) has similarities with the LBM scheme proposed in [18] to solve the advection equation ∂ t ρ + u 0 ∂ x ρ = 0: this point is studied in Section 8.…”

“…and by replacing the first iterate (56) The boundary condition (65) is identical to the one proposed in [6] when α = 1/2 (see (21) and (25) in [6]). Let us also note that (65) is equivalent to…”

“…LBM schemes are now often used to solve many type of Partial Differential Equations (PDEs) system. Among them, we find the heat equation with or without phase change [4,5,6,7], the heat equation with radiative source term [8], the hyperbolic heat equation (also named telegraph equation) with or without radiative source term [9], the Richard equation for porous media [10], the advection-diffusion equation [11,12,13,14,15,16,17], the advection equation [18], the incompressible Navier-Stokes system [19] eventually applied in porous media with heat and mass transfer [20] or in a diphasic situation [21,22] or with a free-surface [23], the Bingham model for viscoplastic flows [24]. Among the reasons which justify the use of LBM schemes, we can cite its algorithmic simplicity, its time explicit nature, its scalability when the algorithm is parallelized.…”

“…In the pure diffusive case (i.e. u = 0), one of these LBM schemes is classical and can be found in [4,6,7,32] (the second one seems to be less classical). As in [29], these LBM schemes are obtained by discretizing a discrete velocity kinetic system whose the fluid limit is (1), this fluid limit being formally obtained with a Chapman-Enskog expansion and with a Hilbert expansion.…”